Large-eddy simulation of non-isothermal flow over a circular cylinder

International Journal of Heat and Mass Transfer 151 (2020) 119426

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Large-eddy simulation of non-isothermal flow over a circular cylinder Sourabh Jogee, B.V.S.S.S. Prasad, Kameswararao Anupindi∗ Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, T.N. 600 036, India

a r t i c l e

i n f o

Article history: Received 23 September 2019 Revised 23 December 2019 Accepted 26 January 2020

Keywords: Large-eddy simulation (LES) Heat transfer Flow over cylinder Vortex shedding Heated cylinder

a b s t r a c t In the present work, flow around a heated circular cylinder, at a Reynolds number of Re = 3900, is investigated using large-eddy simulation (LES). Large differences in temperature, 25 ◦ C, 100 ◦ C, 200 ◦ C, and 300 ◦ C between the cylinder and the oncoming flow are considered, and its effect on the flow and thermal characteristics in the near wake region are studied. The numerical methodology employed is validated for both, the mean and second-order statistics, with the direct numerical simulation (DNS) data available in the literature. The results are analyzed using the mean temperature, velocity, Reynolds stresses, temperature variances, turbulent heat fluxes and energy spectra. The flow and thermal characteristics are studied along the center line in the wake, and in the transverse direction at two locations. The non-isothermal flow characteristics are compared with isothermal flow, to study the effect of temperature on the flow dynamics. Phase-averaging is performed to analyze the regions of turbulence production and convection of heat. It is observed that, the flow characteristics vary non-linearly with the temperature, and the effect is insignificant till a temperature difference of 100 ◦ C, however, beyond this significant effect could be noticed. The effect of temperature difference is prominent in the thermal characteristics for all temperature differences, 25 ◦ C to 300 ◦ C, considered. The transverse component of shear stress fluctuations are observed to be dominant over the stream-wise components at both the locations downstream, thereby enhancing the local mixing of the fluid and hence, the heat transfer. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Non-isothermal flow over a circular cylinder finds applications in heat exchangers, as pin-fins in electronic components, control rods in nuclear reactors, and in the cooling applications in the trailing edge region of gas turbine blades. Owing to the vast applications of engineering interest, and the complex flow and heat transfer dynamics in the turbulent wake, it was studied by several researchers in the past few decades. Both experimental and numerical techniques were used in the past in order to study this problem. Parnaudeau et al. [1] studied the flow over a circular cylinder numerically as well as experimentally. They used LES together with immersed boundary method (IBM) for performing the simulations, and used hot-wire anemometry and particle image velocimetry (PIV) for extracting the experimental data. They focused on obtaining the turbulent statistics as well as power spectra in the wake of the cylinder. Lysenko et al. [2] performed LES at a Re = 3900 and Mach number (Ma) of 0.2. They investigated the influence of two subgrid-scale (SGS) models, using OpenFOAM, and assessed

∗

Corresponding author. E-mail address: [email protected] (K. Anupindi).

https://doi.org/10.1016/j.ijheatmasstransfer.2020.119426 0017-9310/© 2020 Elsevier Ltd. All rights reserved.

the quality of the solution obtained. Meyer et al. [3] assessed the accuracy of conservative immersed interface method (CIIM) using the flow over a cylinder at Re = 3900. The proposed CIIM was able to capture the turbulent wake of this problem with sufficient accuracy. Coming to the experimental studies, the heat transfer from heated circular cylinder placed in a laminar air flow was experimentally investigated by Wang and Trávníˇcek [4]. Their purpose was to determine the reference temperature for the calculation of kinematic viscosity, for describing the forces of convection of the heated bluff body. The same study was extended by Baranyi et al. [5] for different temperatures of the cylinder and the Reynolds number, both experimentally and numerically. They suggested an equation for the temperature dependent coefficient for the definition of reference temperature. An experimental technique, to measure heat transfer from the cylinder surface to the fluid, for a fixed value of wall heat flux was developed by Baughn et al. [6]. The experiments were conducted for three different arrangements, namely, single, tandem and cylinder in the entrance region of the tube bank, for a fixed pitch-ratio and for two sub-critical Re. Their objective was to obtain reliable data with the well-specified boundary conditions. They found that, the influence of Kármán vortex is dominant in the vicinity of the near wake of the cylinder, and it gradually weakens away from the cylinder. Experimental study of variation of momentum and heat transfer associated

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S. Jogee, B.V.S.S.S. Prasad and K. Anupindi / International Journal of Heat and Mass Transfer 151 (2020) 119426

with these vortices was carried out by Matsumura and Antonia [7] and Antonia et al. [8]. Matsumura and Antonia [7] used hot wire anemometry to study the heat transfer contribution from average momentum and heat fluxes. Antonia et al. [8] determined the variation of turbulent Prandtl number, velocity correlations and temperature fluctuations in detail. From their study they concluded that, vortices transport the heat effectively than momentum in the far wake region. The experiments conducted by BUYRUK [9], aim to determine the variation of Nusselt number and pressure coefficient for one tube within a staggered tube bundle and within a row of similar tubes for two different longitudinal and transverse pitch. He also investigated the influence of the blockage ratio for all the arrangements considered. The flow structures and heat transfer phenomenon of each row were explained in detail in that study. The characteristics of the near-surface flow are reflected by unsteady heat transfer, which demand a study of the near vicinity of the body surface. Nakamura and Igarashi [10] performed experiments for Re ranging from 30 0 0 to 15,0 0 0. The cylinder temperature was kept at a fixed value to maintain a temperature difference of 25 ◦ C between the flow and the cylinder. In addition to the aforementioned experimental studies, several researchers have investigated using numerical simulations, in order to understand this flow. The effect of vortex shedding frequency on forced convection heat transfer from bank of circular tubes, in heat exchangers, under cross flow was studied by kumar and Jayavel [11] for different blockage ratios. They confirmed that if the blockage ratio is less than or equal to 3 then the heat transfer enhancement in the heat exchanger does not depend on the flow shedding frequency. The numerical study for the mixed convection heat transfer over cylinder was carried by Salimipour [12]. Their approach was aimed at eliminating vortex shedding with lowest possible drag coefficient, in order to reduce vibration. Alnaka et al. [13] performed an experimental as well as numerical study of the effect of impinging water jet on a heated cylinder, for convection heat transfer on three cylinders arranged in tandem, for 50 0 0  Re  20, 0 0 0. In the simulations the distance between the cylinder and the jet was varied as H/S = 4, 6 and 10, to analyze its effect, but was kept constant in the experiments. Kim and Nakamura [14], studied the instantaneous Nusselt number distribution on the rear surface of the cylinder for three Re = 3900, 10, 000, and 18, 900. The reason for choosing this range of Re is such that, the characteristics of flow and heat transfer change rapidly and therefore make an excellent test case to evaluate the capabilities of LES for such class of flows. The work of [2,14] was combined by Salkhordeh et al. [15]. The computational domain and setup were the same as used by Lysenko et al. [2], however, the flow was non-isothermal and the results were compared for the same parameters as in [14], except the Reynolds number. Witte et al. [16], aimed at identifying the heat transfer response for a pulsating laminar and sub-critical flow across a cylinder using two-dimensional simulations and LES. The two-dimensional simulations employed laminar flow, at Re = 0.4, 4 and 40. Whereas, LES was done for subcritical flow at Re = 3900. In both cases, only half of the domain was simulated using symmetric boundary condition, however the geometry was extended in the third dimension for LES. They presented the heat transfer response for the imposed velocity perturbations. From the foregoing review it is evident that, the studies in the past focused on a single, and a small value of temperature difference between the flow and the cylinder, and therefore temperature was treated as a passive scalar. However, in the real applications, as internal cooling of trailing edges at the initial stages of high pressure gas turbine, will have temperature differences more than 200 ◦ C. This aspect of higher temperature difference between oncoming coolant and cylinder in actual practice is not yet considered in the literature. Only a handful of studies, by Kim and

Nakamura [14] and Salkhordeh et al. [15], can be found which performed LES of non-isothermal flow over a single cylinder, with evaluation limited to the comparison of Nusselt number for temperature difference of 25 ◦ C. Therefore, the aim of the present work is to investigate the flow and thermal characteristics in the near vicinity of cylinder wake for flow over a circular cylinder for a range of temperature differences of 25 ◦ C to 300 ◦ C between the cylinder and oncoming flow. The Reynolds number are small for applications such as turbine blade trailing edge, flows in several low Reynolds number heat exchanger and flow over electronic component. Previous investigations, Parnaudeau et al. [1], Lysenko et al. [2], Meyer et al. [3], Nakamura and Igarashi [10], Kim and Nakamura [14], Witte et al. [16], used Re = 3900 as a typical Reynolds number to represent sub-critical flows. In view of this, the authors have chosen this particular Reynolds number to perform the non-isothermal flow simulations. The remainder of the paper is organized as follows. Section 2 explains the flow domain, meshing strategy and the details of the computational setup. Validation of the solver and the results obtained are explained in Section 3. The conclusions drawn from the present study and directions for future work are identified in Section 4. 2. Numerical methodology In the present section, the LES equations are explained, followed by the computational domain and the mesh which is used. The time-averaging strategy that is employed in the present work is also discussed. 2.1. Large-eddy simulation (LES) equations The governing equations are the unsteady, compressible, filtered mass conservation, Navier-Stokes and energy equations. The filtered mass conservation equation is given as follows,

∂ ρ ∂ ρ uj + = 0, ∂t ∂xj

(1)

where ρ is the fluid density, uj is the velocity vector, t is the time and xj are the spatial coordinates. The Favre-filtered Navier-Stokes equations are given by,

∂ ρ ui ∂ ρ ui uj ∂ σˆi j ∂ p ∂τi j + = − − , ∂t ∂xj ∂ x j ∂ xi ∂ x j

(2)

where p is the fluid pressure, σˆi j is the viscous stress tensor and τ ij is the SGS stress tensor. The Favre-filtered energy equation is given as follows,

∂ ρ Eˆ (∂ ρ Eˆ + p)uj ∂ ui σˆi j ∂ qˆj + − + ∂t ∂xj ∂xj ∂xj  ∂τi j 1 ∂ ( pu j − pu j ) =− − uj , γ −1 ∂xj ∂xj

(3)

where Eˆ is the total energy, γ is the specific heat ratio of the fluid, qˆ is the heat flux vector. In the above equations, the filtering operation is denoted by an over-line (φ ), where φ is any arbitrary physical property. Another filtering operation represented by tilde ( φ ), refers the Favre-filtering, Garnier et al. [17], that is used in i = ρ ui /ρ . The thermodycompressible flow, and is defined as u namic variables are connected using the equation of state, given as, p = ρ R T , where R is the gas constant, and  T is the filtered temperature. The total energy, can be evaluated using the following equation,

ρ Eˆ =

p

γ −1

+

1 ρ ui ui , 2

(4)

S. Jogee, B.V.S.S.S. Prasad and K. Anupindi / International Journal of Heat and Mass Transfer 151 (2020) 119426

and the viscous stresses occurring in the filtered Navier-Stokes Eq. (2) can be expressed as,

    ∂ ui ∂ uj ∂ uk 2   σˆi j = μ + − μ δ , ∂ x j ∂ xi 3 ∂ xk i j

(5)

further, the heat flux vector occurring in the filtered energy Eq. (3) can be expressed as follows,

qˆi j = −

C p ∂  μ T , Pr ∂ x j

(6)

where Pr represents the molecular Prandtl number. In the present study air is considered as the working fluid for which the specific heat ratio γ = C p /Cv = 1.4, and the value of Pr is taken as 0.71. A constant value of 0.85 is considered for the turbulent Prandtl number (Prt ), as suggested by Kader and Yaglom [20]. The SGS stress tensor, that arises as a result of the non-linear velocity terms, can be expressed as follows,





i uj . τi j = ρ u iu j − u

(7)

The SGS stress term, τ ij , needs to be modeled in order to close the system of the filtered equations. Using eddy-viscosity hypothesis, the SGS stress tensor can be expressed as,



1 1 τi j − τkk δi j = −2μSGS Si j − S kk δi j 3

3

(8)

where S i j is the filtered strain rate tensor that can be obtained from the resolved scales, and can be expressed as follows, S˜i j = 1 2



∂ uj ∂ ui ∂ x j + ∂ xi



and the SGS viscosity, μSGS can be evaluated us-

ing Boussinesq hypothesis as, μSGS = ρ (Cs )2 |S˜|, where the model constant Cs is flow dependent and varies from 0.065 to 0.25. The filter width ( ) depends on  the grid spacing and is given by  1/3 = x y z and | S| = 2S S is the second invariant of the ij ij

strain rate tensor. For compressible flows, Yoshizawa [18] proposed a closure for the SGS kinetic energy, τ kk , as follows,

τkk = 2ρCI 2 |S˜|2 ,

(9)

where the model constant, CI , is dynamically obtained for the dynamic SGS models. In case of a dynamic model, instead of using Eq. (9), the transport equation for SGS kinetic energy (k) is solved, given as follows,



∂ ρ k ∂ ρ uj k ∂ qi ∂τi j u˜i ∂ ∂k  + = −R + − τi j S i j + μ˜ ∂t ∂xj ∂xj ∂xj ∂xj ∂xj ∂ fi + − s − c + P ∂xj

(10)

where τ ij and qj are the SGS stress and SGS heat flux defined by Eqs. (7) and (6) respectively, and the remaining terms are defined as

fi =

 1 1  j − u ρ u i ui u i ui u j + 2 3

  ∂u ∂ u uj k , μu j k − μ ∂ xk ∂ xk

∂ ui ∂ ui ∂ ui ∂ ui −μ ˜ , ∂xj ∂xj ∂xj ∂xj    2 2  1 ∂ uk ∂ uk 

c = μ −μ , 3 ∂ xk ∂ xk

s = μ

P=p

∂ uk ∂ u − p k, ∂ xk ∂ xk

where the term fi is the transport term that contains, triple velocity correlation and dilatational diffusion, s is the SGS dissipation term, c is the dilatational dissipation, and P is the pressure dilatation term. These terms are to be modeled in order to close the SGS k equation. The details of modeling can be found in the article by Chai and Mahesh [19].

3

Table 1 Details of the domain extents, grid resolution and the type of grid used by several researchers for simulating flow over a circular cylinder at Re = 3900. Contributors

T

Lx

Ly

Lz

S

Mittal and Moin Breuer, Case D3 Kravchenko and Moin Franke and Frank Alkishriwi Paranaudeau et al. Mani et al. Meyer et al. Lysenko Dmitry A. Current

C O O O C H O O-H O H

36 × D 30 × D 60 × D 30 × D 25 × D 20 × D 70 × D 30 × D 50 × D 25 × D

50 × D 30 × D 60 × D 20 × D 15 × D 20 × D 70 × D 20 × D 50 × D 20 × D

π ×D π ×D 2π × D π ×D

401 × 120 × 48 165 × 165 × 64 205 × 185 × 48 185 × 193 × 33 105 × 273 × 49 961 × 960 × 48 288 × 396 × 48 6 × 106 300 × 300 × 64 430 × 170 × 38

1×D

π ×D π ×D 4×D

π ×D π ×D

2.2. Details of the geometry, mesh and computational setup Fig. 1 shows the schematic of the computational domain used in the present study, with the boundary conditions. The origin of the domain is at the center of the cylinder, with a diameter D, placed across the air stream. The inlet to the domain is kept at a distance of 5D upstream, and the outlet at 20D downstream of the center of the cylinder. In the transverse direction, the domain is extended for 10D on either side. Wissink and Rodi [21] performed a direct numerical simulations (DNS) of incompressible flow around a circular cylinder at Re = 3300. In the study they found that, even a span of 8D is not sufficient to capture the largest span-wise scales of motion. In the same study, they also mentioned that, the span-wise extent has a very small effect on the turbulent statistics, and therefore recommended smaller spanwise width. Therefore, in the present work, a span-wise width of π D, consistent with the other studies, is chosen. Table 1 presents the details of the computational setups used by several researchers in their respective studies. This table is reproduced from [2], and updated with the data related to the present work. From this table it can be noted that, most of the works in the past used a span-wise width of π D except [3,22] which used larger span-wise widths. All the simulations in the present work are performed using OpenFOAM [23]. Velocity boundary condition at the inlet to the domain defined with a uniform Dirichlet velocity field, and outlet is defined with zero gradient. A homogeneous pressure gradient boundary condition is applied at the inlet. At the outlet of the domain an advective boundary condition is used in order to avoid pressure reflections into the domain. A no-slip, no-penetration boundary condition is applied for velocity on the surface of the cylinder. The surface temperature of the cylinder is set to a constant value. The temperature of incoming fluid is kept constant, and the cylinder temperature is varied, to get the desired temperature differences of 25 ◦ C, 100 ◦ C, 200 ◦ C, and 300 ◦ C between the two. Periodic boundary conditions are applied in the span-wise direction. Symmetry boundary conditions are applied on the top and bottom boundaries as shown in Fig. 1. From Table 1 it can be noted that, most of the studies utilized a structured O- and C-type grids. Only few works by Parnaudeau et al. [1], Ma et al. [24] and Wornom et al. [25] employed a H-type grid. The simulations performed by Meyer et al. [3] and Wornom et al. [25] employed unstructured grids. The average number of cells used in the LES studies is about 4 million, with mesh refinement near the surface of the cylinder. The simulations performed by Parnaudeau et al. [1] used highest number of cells, about 45 million cells approximately. In the present study, a rectangular domain is considered with structured H-type grid. The domain is discretized using 430 × 170 × 38 cells in stream-wise, transverse and span-wise direction respectively. Fig. 2 shows the discretized computational domain for 2.06 million cells on the central xy-plane.

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Fig. 1. Schematic showing the computational domain and the boundary conditions.

Fig. 2. The mesh used in present simulation on the central xy-plane; the figure in the inset shows a close-up view of the near cylinder mesh.

A wall-resolved approach is used in the present LES studies, and therefore the mesh is refined near the surface of the cylinder with the centroid of the first cell away from the wall to be less than one, y+ < 1. The near cylinder mesh refinement can be seen in the inset of Fig. 2. The solver used for the present simulations is buoyantPimpleFoam, which is a transient solver for buoyant, turbulent flow of compressible fluids, and uses PIMPLE algorithm (combination of PISO and SIMPLE algorithms) in order to obtain a solution. The temporal terms are discretized using a second-order accurate implicit scheme, and second-order accurate central differencing

schemes are used for evaluating all the spatial derivative terms. The dynamic k-equation model is used as the SGS model following the studies of Kim et al. [26]. 2.3. Time-averaging strategy and formulations In the present simulations for Re = 3900, have a turbulent wake. In order to obtain statistical quantities of interest, these turbulent fields have to be suitably averaged. One way to obtain a suitable time period for averaging is to look at time-scales of the problem. One such time scale, that can be readily identified

S. Jogee, B.V.S.S.S. Prasad and K. Anupindi / International Journal of Heat and Mass Transfer 151 (2020) 119426

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Table 2 Comparison of bubble length and minimum velocity in the wake center line for indicated temperature difference between the cylinder and the flow. Temperature

Iso-thermal 25 ◦ C 100 ◦ C 200 ◦ C 300 ◦ C

Bubble length (L/D)

umin /U∞ (at Location x/D)

% change L/D

umin /U∞ , (x/D)

2.18 2.24 2.26 2.46 2.54

−0.307 −0.301 −0.305 −0.267 −0.254

– 2.75 3.67 12.84 16.51

– 1.95 (3.65) 0.65 (4.87) 13.03 (17.07) 17.26 (20.73)

(1.64) (1.7) (1.72) (1.92) (1.98)

Table 3 Comparison of change in peak values and shift for turbulent statistics for different temperature. Temperature

Iso-thermal 25 ◦ C 100 ◦ C 200 ◦ C 300 ◦ C

2 u u /U∞

TKE (m2 /s2 )

2 v v /U∞

Value

x/D

% Change

Value

x/D

% Change

Value

x/D

% Change

7.544 7.527 7.476 7.35 7.172

2.24 2.24 2.24 2.5 2.58

– 0 0 11.6 15.18

0.109 0.1011 0.097 0.107 0.096

2.24 2.16 2.16 2.42 2.48

– 3.57 3.57 8.04 10.72

0.366 0.383 0.38 0.364 0.359

2.32 2.36 2.3 2.58 2.66

– 1.72 0.86 11.2 14.66

Fig. 3. Transverse variation of normalized mean stream-wise velocity, u/U∞ .

2 . Fig. 4. Transverse variation of mean normal stress, u u /U∞

formula relating St and Re is given by the equation in the present problem is the time-period or frequency of vortex shedding. The frequency of vortex shedding, f, from the cylinder is determined using dimensionless Strouhal number, St, which is useful for analyzing oscillating unsteady fluid flow. An empirical



St = 0.198 × 1 −



19.7 . Re

(11)

The above equation could be used to obtain St for a given Re, and from the definition of St = f D/U∞ , the frequency and therefore the

Fig. 5. Stream lines and lines over which data is collected in transverse direction for T = 300 ◦ C.

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Fig. 6. Mean stream-wise velocity, u/U∞ .

Fig. 7. Axial centerline variation of mean turbulent kinetic energy (TKE) in m2 /s2 for the indicated non-isothermal cases.

time-period of oscillations could be obtained. Franke and Frank [27] obtained a non-dimensional time-scale given by,

t =

t × U∞ D

(12)

where t is the total running time of the simulation and U∞ is free-stream velocity. Using this non-dimensional time-scale they concluded that, the averaging time needs to be larger than t = 200 (or 40 cycles) are necessary to obtain a converged mean flow field. The experimental results published by Parnaudeau et al. [1] considered 60 vortex shedding cycles whereas, computational results of [2] considered 150 cycles. For a quantitative validation, present simulation is performed for t = 690, which gives ∼ 130 number of shedding cycles. The initial transient time, suggested by Parnaudeau et al. [1], is given as follows,

t=

150 × D . U∞

(13)

Therefore, in the present study the turbulent statistics are collected after excluding an initial transient time of t ∗ = 150. Throughout the paper, the time-average fields are denoted by φ, and the fluctuations are represented by φ  , where φ is any arbitrary physical quantity.

2 2 (b) v v /U∞ . Fig. 8. Axial centerline variation of (a) u u /U∞

3. Results and discussion In this section, the grid sensitivity study and the validation are presented. Further, the results obtained using the present simulations are discussed using the flow and thermal characteristics. The mean and second-order statistics are presented, both in the stream-wise and in the transverse direction. The power spectra of velocity and temperature fluctuations are presented, in order to understand the energy cascade process. The results are normalized using the free-stream velocity, U∞ , and temperature, T∞ . 3.1. Grid sensitivity test and validation The present study focuses on the near-wake dynamics of flow and heat transfer. In order to accurately resolve the physics, the near region of the surface of the cylinder is refined, as shown in Fig. 2, with an expansion ratio of 10, the ratio of largest cell-height to the smallest cell-height. A grid sensitivity study is performed to ensure the results obtained converge as the mesh is refined. The results obtained from different mesh resolutions are analyzed at two different locations viz., x/D = 1.06 and 1.54, in the wake of the cylinder. The grid sensitivity test is conducted for Re = 3900 and for the isothermal flow situation, that is for the same temperature

S. Jogee, B.V.S.S.S. Prasad and K. Anupindi / International Journal of Heat and Mass Transfer 151 (2020) 119426

Fig. 9. Mean velocity variation with different temperature difference at y/D = 1.06 and 1.54. (a) In stream-wise direction. (b) In transverse direction.

of the cylinder and the oncoming flow, as the DNS data is available only for the isothermal case. In order to validate the results, the DNS data, simulate by Ma et al. [24], and the experimental results obtained by Parnaudeau et al. [1], for the isothermal flow over a cylinder, are used. Fig. 3 shows the transverse variation of normalized mean streamwise velocity, at indicated locations together with the reference data. Three different grid resolutions, such as, 1.52, 2.06 and 2.52 million cells are used in order to evaluate the sensitivity of the grid on the solution obtained. From this figure, it can be observed that, the mean velocity profile obtained using a mesh of 1.52 million cells shows deviation from the reference data, however, the profiles obtained using 2.06 million and 2.52 million cells, show a good match with the DNS and the experimental data. In fact, the match is found to be better when compared with the LES results of [2] with the reference data. It should be noted that the present mesh utilizes 2.06 million cells which is approximately 2.8 times smaller compared to the mesh resolution of [2], which further justifies the accuracy of the method employed. In addition to the variation of mean velocity, the second-order statistics are also compared with the reference data. The transverse variation of 2 is shown in Fig. 4. From this figure also, normal stress, u u /U∞

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Fig. 10. Velocity fluctuations at different temperature difference at x/D = 1.06 and 1.54. (a) In stream-wise direction. (b) In transverse direction.

Fig. 11. Variation of turbulent kinetic energy (TKE) in m2 /s2 at x/D = 1.06 and 1.54.

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Fig. 12. Iso-surface of instantaneous λ2 = −10 s−2 colored by instantaneous velocity magnitude.

Fig. 13. Contours of turbulence intensity, for a temperature difference of 300 ◦ C.

the results obtained using 2.06 million cells and 2.52 million cells almost overlap over the entire range and show a good match with both the DNS and the experimental reference data. The comparison of present results using Figs. 3 and 4 shows that, there is no difference in the prediction of mean stream-wise velocity and fluctuations at both the locations, between the cases with 2.06 million and 2.52 million cells. Therefore, as there are no appreciable changes in the results obtained using 2.52 million cells case, the 2.06 million cell case is used in order to perform further simulations, involving heat transfer. The present simulations are performed considering constant properties of the air. However, in order to ascertain the effect of variable properties, further simulations were run. From these simulations it was noted that there are no differences in results between the two. This is because the maximum increase in temperature of the fluid is observed to be only about 30% and 20% at

x/D = 1.06 and 1.54 respectively, which is not sufficient to cause an appreciable change in the properties of the fluid. 3.2. Flow characteristics In this section the flow characteristics, such as the mean and RMS profiles are studied along the central axial line, and along two vertical lines in the wake of the cylinder, at x/D = 1.06 and 1.54. The differences in flow dynamics obtained, as a result of varying the temperature difference, are compared with the results obtained using the isothermal case. Variation in the stream-wise direction The variation of mean stream-wise velocity for different temperature differences are compared with the isothermal case, and are shown in Fig. 6. This figure shows that the length of the

S. Jogee, B.V.S.S.S. Prasad and K. Anupindi / International Journal of Heat and Mass Transfer 151 (2020) 119426

Fig. 14. Axial variation of normalized temperature variance, T T /T∞ .

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Fig. 16. Transverse variation of mean temperature, T/T∞ , at the indicated x/D locations.

Fig. 15. Turbulent heat flux in stream-wise direction, u T /U∞ T∞ . Fig. 17. Transverse variation of normalized temperature variance, T T /T∞ .

recirculation bubble, L, a point where velocity changes from negative to positive, increases as the temperature difference is increased. The magnitude for the length of recirculation bubble and the minimum velocities in this region are presented in Table 2. The percentage change in x/D location is calculated with respect to the isothermal case. For example, for the flow bubble length, the change is calculated as, %change = (Li − Lisothermal )/Lisothermal . This variation of the flow parameters is non-linear as the variation between 25 ◦ C to 100 ◦ C and 200 ◦ C to 300 ◦ C are small compared to changes observed between 10 0 ◦ C to 20 0 ◦ C. Table 3 shows the highest values of TKE, u 2 /U∞ , v 2 /U∞ and their percentage change in the shift of the peak turbulent statistical quantities, over the centerline in the wake. It can be noted that, the change in the magnitude of these parameters is not appreciable, however, the changes in the locations where the peak values occur is noticeable between 100 ◦ C and 200 ◦ C. The profiles of second-orders statistics, for different temperature differences, are observed to collapse on single value beyond x/D = 5. The reason for this behavior can be explained by looking at the plot of turbulent kinetic energy (TKE), as shown in Fig. 7 and the contours for the same, as shown in Fig. 20a for a temperature difference of 300 ◦ C. As x/D increases, the velocity fluctuation initially increases till they reach a peak value and then weaken. The location of the peak TKE shifts away from the cylinder center with an increase in the temperature difference. Therefore, it can be concluded that, the effect of increasing the temperature difference is

to shift the peak of the TKE away from the cylinder center and the magnitude of peak is inversely correlated with the temperature difference. The axial centerline profiles of u 2  and v 2  is shown in Fig. 8(a) and (b) respectively. It can be observed from these figures that, Because of the oscillation of the flow in transverse direction, v 2  are three times higher than u 2 , however, qualitatively their axial variation looks similar. The higher values of v 2  helps in enhanced mixing of the fluid locally thereby it enhances the heat transfer. The profiles of u 2  for different temperature differences are all lower in magnitude when compared with the profiles of the isothermal case. However, the profiles of v 2  for the nonisothermal cases are not appreciably different from the isothermal case. This further suggests that the temperature difference plays a prominent role in the stream-wise component of stress in comparison to the transverse component of the stresses. Variation in the transverse direction Fig. 9 (a) and (b) show the profiles of mean stream-wise and transverse velocity for the transverse direction at the indicated x/D locations respectively. The effect of temperature on u is minimal, however, its effect on the profile of v is appreciable at x/D = 1.06. The effect of temperature on both stream-wise and transverse components of velocity can be noted to be dominating at a downstream location of x/D = 1.54. In the vicinity of

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Fig. 18. Transverse variation of turbulent heat flux (a) stream-wise component (b) transverse component at indicated locations.

Fig. 19. Variation of Nusselt number, Nu, over cylinder surface.

cylinder, −0.5  y/D  0.5, the profiles of u/U∞ at x/D = 1.06 are flat and no change can be be observe at any temperature difference in comparison to the isothermal case. At x/D = 1.54 the profiles are in the shape of the letter ‘V’ for the isothermal and lower temperature difference cases, whereas a clear U-shaped profiles can be observed for higher temperature difference cases. Unlike the profiles of u, the nature of v is qualitatively the same at both the x/D locations downstream. For the profile of v small change could be observed at x/D = 1.06 with the temperature difference, however, the effects are more pronounced as one moves further downstream. From the variation in the transverse component of velocity, it appears that the extreme peaks of these profiles come closer to each other as the temperature difference is increases. At the location y/D = 0, all the profiles attain a zero magnitude because, at this point the upper and lower recirculation bubbles merge and hence there is no transverse velocity, as can be seen in the streamline contours in Fig. 5. The variation of mean velocities is also observed to be higher for the temperature differences of 100 ◦ C to 200 ◦ C, consistent with the earlier observations. Fig. 10 (a) and (b) show the variation of the normal components of stresses in the stream-wise and transverse directions respectively. Comparison of these two figures shows that, in near wake region, that is at x/D = 1.06, the u 2  is dominant, nearly 2.5 times compared to v 2 . However, the profiles of v 2  have a

dome-shape in the central region, and this effect can be noticed on the profiles of TKE as shown in Fig. 11. The magnitudes of u 2  and v 2  are of comparable oder further downstream at the location x/D = 1.54. At the centerline, the profiles of u 2  have much lower values compared to the profiles of v 2 . This is in contrast to the profiles of v, which attain zero values in the centerline, as noted in Fig. 9b. It can be noted that the in the centerline region, at x/D = 1.54 the mean transverse velocity is zero, however, the transverse velocity fluctuations are non-zero. Next, we analyze the variation of TKE in the wake of the cylinder. It can be noticed from Fig. 11 that, the effects of turbulence can be felt only in the range of −1  y/D  1, beyond which the flow is laminar. However, the width of this region increases as one moves downstream, as can be noted from the contours of TKE in Fig. 20a. As observed in Fig. 7, the values of TKE initially increase and then decrease as one moves away from the cylinder, therefore the magnitude of TKE in Fig. 11, at location x/D = 1.54 is higher compared to at the location x/D = 1.06. At both the locations, the magnitude of the peak of the TKE decreases with an increase in the temperature difference, however, the profiles are not qualitatively similar at both the locations. At x/D = 1.06 the TKE profiles are flat in the central region bounded by two sharp peaks, however,

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Fig. 20. Contours of turbulent kinetic energy in m2 /s2 (a) time-averaged and (b) phase-averaged. Contours are presented for a temperature difference of 300 ◦ C.

at x/D = 1.54 the peaks of the profiles have diffused and have also came closer to the axis. Further, through can be noticed between two peaks at x/D = 1.54. The iso-surface contours of instantaneous λ2 = −10 s−2 are shown in Fig. 12. From this figure it can be noted that the shear layer that is attached to the cylinder breaks down about twodiameters downstream of the cylinder. Further, it can be noted that the vortical structures are generated near the aft of the cylinder and extend in the stream wise direction. A number of small size structures can be observed in the immediate wake of the cylinder. The structures that emanate from the top and bottom of the cylinder, owing to vortex shedding, grow as they convect downstream. From the above discussion, it can be concluded that all the profiles, Figs. 9–11, are symmetric about the center line and the changes are not appreciable when compared with the isothermal flow, until a temperature difference of 100 ◦ C. Drastic changes in the profiles, in comparison to the isothermal case, can be observed beyond a temperature difference of 100 ◦ C until 200 ◦ C and after which the changes subside over the range of 200 ◦ C to 300 ◦ C.

3.3. Thermal characteristics In this section, heat transfer characteristics are discussed at the same locations of x/D = 1.06 and x/D = 1.54. The variations in mean temperature, turbulent heat flux and temperature fluctuations are used in order to illustrate the effect of the temperature difference. Variation in the stream-wise direction The axial variation of normalized temperature fluctuations are shown in Fig. 14. From this figure, it can be noted that the temperature fluctuations are highest very near to the rear stagnation point, which is the point where the upper and lower recirculation bubbles separate, as shown in fig. 5, results in increase of turbulence intensity. The contours of turbulence intensity are shown in Fig. 13. The contours show a high intensity of turbulence at the rear stagnation point and thus high temperature variance. This can also be observed in conjunction with TKE. The zoomed in view in Fig. 7 shows a drastic increase of TKE very near to rear

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Fig. 21. Contours of stream-wise velocity turbulent heat flux for temperature difference of 300 ◦ C, (a) time averaged (b) phase averaged.

stagnation point and hence this results in high temperature fluctuation at this location. Although the values of TKE decrease with an increase in the temperature difference, it is noted that higher temperature difference result in higher temperature variances. The temperature fluctuations are observed to drop suddenly in the range of 1 ≤ x/D ≤ 2, and are found to increase again towards the end of the recirculation zone. Thereafter due to the mixing of cold fluid, the temperature variances decrease and approach towards zero. As expected, the value of x/D, where this variance completely disappears, increases with the temperature. The axial variation of turbulent heat flux, in stream-wise direction is shown in Fig. 15. The values of heat flux increase with an increase in the temperature difference and attain peak values around the same location, which is consistent with the temperature variances. Thereafter, the values of turbulent heat fluxes decrease gradually and just before the end of recirculation zone, they decreases drastically and reach a minimum negative value. Both, the highest and lowest, attained by turbulent heat fluxes are associated with the highest temperature difference case. The nature of the curves indicates that, the fluid gains heat flux, indicated

by positive values, from cylinder initially at a high rate and then it decreases slowly just before the end of the recirculation zone, after one point (which is varies with the temperature difference) the turbulent heat flux decreases drastically, and reaches a value of zero. At the end of the recirculation zone, the hot fluid starts diffusing the heat to the cold fluid, presented by negative values in the curve. This negative value is because the cold fluid is transported in the negative x-direction. Upon observing the upper half cycle in Fig. 25 (from t = 1.02 ms to 5.08 ms) shows that, the cold fluid is convected in the negative x-direction and the direction of rotation of the small vortex is in the clock-wise direction. Through diffusion, the fluid just besides the recirculation zone is warmer compared to the fluid away from cylinder resulting in a lower diffusion of heat, however, as the fluid advances downstream this rate increases, and reaches a value of zero asymptotically. Variation in the transverse direction The profile of the mean temperature in the transverse direction is shown in Fig. 16. The nature of profile for T at both the locations, x/D = 1.06 and 1.54, is nearly the same except that, in

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Fig. 22. Contours of transverse turbulent heat flux for temperature difference of 300 ◦ C, (a) time averaged (b) phase averaged.

between the two peaks, near the cylinder the profile has a convex shape, and the one at the downstream location has a concave shape in higher temperature difference cases. As the TKE is higher at x/D = 1.54 compared to at the location 1.06, as is noted in Fig. 7, the mixing of fluids results in a decrease in the mean temperature for x/D = 1.54 and this results in concave shape of the profile. Although the peak increases with an increase in the temperature difference, the location where the maximum occurs remains fixed at the location |y/D| = 0.624 and 0.584 for x/D = 1.06 and 1.54 respectively. These locations of the peaks in terms of y coordinates exactly equal the height of the recirculation zone shown in Fig. 5. Similar to the profile of mean temperature, the profile of temperature variance also displays two peaks as shown in Fig. 17. In addition, several local peaks can be observed at x/D = 1.06 for the higher temperature difference cases. The reason for this can be explained with the help of the stream lines, plotted in Fig. 5. The first two peaks occur at the intersection of the recirculation zone and free stream flow. After the second peak, temperature variance profiles show a downward trend till |y/D| = 0.5, and again shown an upward trend till the center point y/D = 0. It can be observed

from the mean streamlines plot, shown in Fig. 5, that the locations where T T  is the largest, at this point the two streamlines from the upper and lower halves get separated and result in an increase in the magnitude of fluctuations. However, at the location x/D = 1.54 the upper and lower streamlines are parallel to each other, thereby cause smaller fluctuations. Further, this is the reason for the concave shape of the profiles at this locations for higher temperature difference cases. At lower temperature difference the effect of temperature is not felt at this location and hence a flat profile is observed in Figs. 16, and 17. Fig. 18 (a) and (b) show the variation of turbulent heat flux in the transverse direction, and contours for the same respectively. Fig. 18(a) shows that the highest positive and lowest negative values, that is maximum heat flux gain and maximum heat flux loss respectively, are at the inside and the outside edges of the recirculation zone for x/D = 1.06. A second maximum occurs at y/D = 0, which is the point very close to the separation of the upper and lower recirculation bubbles. This causes an increase in the turbulence level and thereby increases the heat flux. As the temperature difference increases, the magnitude of turbulent heat flux

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Fig. 23. Contours of the convection operator of the heat flux budget equation, u T , for a temperature difference of 300 ◦ C for the indicated phase angles, φ .

diffusion increases. With an increase in the temperature difference, the point of maximum gain in heat flux remains at the location |y/D| = 0.576, whereas the point of maximum diffusion shifts away from the cylinder. This shift of the point of maximum diffusion is because of the increase in area of the high temperature fluid. From the figure, at x/D = 1.54, a lowest value occurs at the outside edge, however, the highest value attained lies just near the centerline and this location gets closer to centerline with an increase in the temperature difference. Although, the location of maximum heat loss and gain are different for the two x/D locations, the magnitude of maximum heat loss at both the locations is almost the same. Fig. 18 b shows the variation of turbulent heat flux in the transverse direction, and the contours for the same are shown in Fig. 22a. A pair of maximum and minimum can be observed either of the central point, y/D = 0. In this figure, and all the profiles cross the axis at y/D = 0. The qualitative nature of the profiles at both the locations is the same, however, the magnitudes are different. Because of the presence of the recirculation zone, just behind the cylinder, the cold fluid does not mix with the hot fluid which causes a very low turbulent heat flux in the transverse direction, as shown in Fig. 22a. However, at the end of the recirculation zone, because of vortex oscillation in the transverse direction, the turbulent heat flux in this direction increases. This is the reason for v T  to attain smaller values at x/D = 1.06 compared to those at x/D = 1.54. Till now the effect of temperature over the flow characteristics is discussed in details, but the heat transfer from the cylinder is

also an important aspect, and presented in terms of average and fluctuating Nusselt number in Fig. 19. No difference in the prediction of average as well as fluctuating Nusselt number can be observe till 90◦ due to the boundary layer being laminar. The value of average Nusselt number at the point of flow separation, i.e., minimum value of Nu, is the same for all the temperature differences but the angle and hence the point of flow separation is delayed with the increase of temperature difference. The sudden increase in fluctuating Nusselt number, as can be seen from Fig. 19b, is due to the increase in turbulence after the separation. The fluctuating Nusselt number continues to increase, after flow separation, as one moves towards the rear stagnation point. The decrease in fluctuating Nusselt number with temperature is due to an increase in the viscosity of air with temperature. From the above discussion, it can be concluded that, with an increase in the temperature difference, the variation of all the thermal characteristics are linear, compared to the flow dynamics. The difference in magnitude between the profiles even at lower temperature difference is appreciable, which was not the case for the flow parameters till 100 ◦ C. Similar to the flow parameters, the heat transfer characteristics are also symmetric about the centerline. Phase-averaged thermal characteristics In this section the phase-averaged thermal characteristics are discussed to gain more insights into the regions of turbulence production and convection of heat and regions where the fluid is getting heated by the cylinder. One shedding cycle period is cal-

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culated from equation 11, and this period is divided in five equal intervals for data sampling. A total of twenty vortex shedding cycles are considered in order to calculate phase-averaging. The location of turbulence production is determined by looking at regions of maximum turbulent kinetic energy (TKE). A comparison of time-averaged and phase-averaged TKE is presented in Fig. 20. This figure shows that turbulence is generated near the end of the recirculation zone and the corresponding coherent structures could be observed in Fig. 12. A comparison of these figures clearly indicates the locations of maximum TKE to be the same from both time-averaged and phase-averaged results. Contours are presented only for a phase angle of φ = 288◦ . However, at any phase the contours obtained are noted to be similar between the two-averaging procedures considered. This indicates that the regions where maximum TKE values are obtained do not change much with the time. The time and phase-averaged contours for the turbulent heat flux in the stream wise, and transverse directions are shown in Figs. 21 and 22 respectively. From Fig. 21 it can be observed that, the fluid is being heated in the recirculation zone and further downstream heat is diffused to the free stream flow. A comparison of both these figures shows a minute change in the heat flux up to x/D = 1.54, thereafter the time-averaged results show a set of vortices, whereas the phase-averaged contours show a set of convecting vortices thereby causing differences further downstream. The contours for the convection term of the turbulent heat flux

  ∂ ui T   ∂ x , Naqavi et al. [29], for different

budget equation, C = u j

j

phases are presented in Fig. 23. In this figure, contours of red indicate a gain of heat flux and contours of blue indicate a loss. From this figure it is clear that, in the near wake, maximum heat gain is at the inside edge of the recirculation and heat loss is near the outside edge. This is in accordance with the previous observation made in the Fig. 18. Away from the cylinder, this gain-loss pattern is noted to be reversed. This is because, the fluid penetrates in the transverse direction due to vortex shedding, and a vortex formed in the previous cycle gets stretched, and convected away from the cylinder, for example see Fig. 25 for t = 8.1 ms and t = 9.1 ms. As the fluid convects heat gets diffused to the free stream flow, which is at a lower temperature.

Fig. 24. Variation of instantaneous transverse velocity with time. Figure (b) shows zoom view from 1 s to 1.1 s.

3.4. Instantaneous flow and energy spectrum analysis In this section, the details of instantaneous flow field and the turbulent energy cascade process are discussed using energy spectrum analysis. The instantaneous flow characteristics are discussed in terms of vorticity for vortex shedding of one cycle. Instantaneous flow field analysis The variation of instantaneous transverse velocity with respect to time is shown in Fig. 24a, and part of which is shown in Fig. 24b for better visualization. Values are obtained by inserting probe at the location x/D = 3 and y/D = 0. The distance between two successive peaks, or valley, shows the period of one vortex-shedding cycle, which is equal to 0.01 s and can also be cross verify with Eq. (11). The fluctuations in the curve show that, the flow has multiple frequencies at this point; hence, the magnitude of the peaks are vary from cycle to cycle. The sequence of instantaneous vorticity contours for one shedding cycle are shown in Fig. 25, for a temperature difference of 300 ◦ C. The vortex shedding is clearly visible with small vortices within the larger vortical structures, emanating from the sides of the cylinder. The vortices with high strength can be observed from x/D = 0.015, due to short recirculation region, Yeon et al. [28]. The comparison of structural changes associated with time advancement of the vortex is visible in these contours. The change in phase

of the cycle can be seen at t = 5.08 ms, after which, the present vortex, dark red spot, starts to weaken and begins formation of a new vortex. Energy spectrum analysis Next, the energy spectrum of the flow field is analyzed. In order to obtain the energy spectrum, the data is collected at, x/D = 1.54, y/D = 0 and z/D = 1.5708. The data is collected for a time span of U∞ t/D ≈ 140. The frequency is non-dimensionalized using Strouhal number, and power spectrum density by squaring the absolute value of FFT output and scaling it by a factor equal to (1/number of samples) × (1/Fs), where Fs is the sampling frequency. Following discussion is presented for a temperature difference of 25 ◦ C. The normalized power spectra obtained for the stream-wise and transverse velocities at x/D = 1.54 is shown in Fig. 26. The peak for the frequency of velocity fluctuations can be observed at the vortex shedding frequency, calculated with the help of Eq. (11). Comparison of figures shows that, the stream-wise velocity has more fluctuations compared to the transverse velocity with an order of magnitude. The energy spectra for temperature fluctuations at both the locations are shown in frames (a) and (b) of Fig. 27. The spectra show that, the diffusion of energy is higher at x/D = 3 compared to

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Fig. 25. A set of instantaneous vorticity fields for one cycle of vortex-shedding for the temperature difference of 300 ◦ C.

Fig. 26. Energy spectrum at x/D = 1.54 for velocity fluctuations for the temperature difference of 25 ◦ C in (a) stream-wise direction (b) transverse direction.

Fig. 27. Energy spectrum for temperature fluctuations for the temperature difference of 25 ◦ C at (a) x/D = 1.54 (b) x/D = 3.

x/D = 1.54, because, the location x/D = 1.54 lies within the recirculation zone and the other one lies outside of it. At x/D = 3 cold fluid gets mixed, as can be seen in Fig. 5, causing higher diffusion of temperature at that point, Fig. 27b. The fluid at x/D = 1.54 is warmer compared to fluid at x/D = 3 resulting in gain of energy and very small diffusion of the temperature, this can be observed in Fig. 27a. From the foregoing discussion it can be concluded that, the vortex shedding process is observed beyond x/D = 0.015 due to the presence of a small recirculation region. From the energy spectra it is clear that present simulations are able to capture the inertial sub-range correctly, and the diffusion rate of transverse velocity;

as well as the temperature inside the recirculation zone is lower compared to that outside this zone. 4. Conclusions and future work In the present work, the flow over a heated circular cylinder is simulated for a range of temperature differences varying from 25 ◦ C to 300 ◦ C for a Re = 3900. The temperature of the incoming flow is kept constant and the temperature of the cylinder is varied to obtain the required temperature differences. The numerical method used is thoroughly validated with the experimental and numerical simulations from the literature, for the isothermal case.

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A good match is found between the present results and the reference data for both mean and second-order statistics. Thereafter, the effect of temperature differences on the flow and heat transfer characteristics is analyzed. The following conclusions can be drawn from the present study: •

•

•

•

•

•

•

•

•

The effect of temperature difference on the flow characteristics is found to be nonlinear. Small changes in the second-order statistics of the flow are observed for temperature differences up to 100 ◦ C, however, prominent effect is found for the temperature difference in the range of 100 ◦ C to 200 ◦ C. The effect of an increase in temperature difference can be perceived only up to x/D = 5, in the stream-wise direction, and in the range −1  y/D  1, in the transverse direction; beyond which all the profiles collapse on to a single curve. However, downstream of the cylinder, the range for y/D where the effect can be felt widens. A comparison of the turbulent stress components, in streamwise and transverse directions, reveals that the fluctuations in the transverse component are dominant over the stream-wise components at every location, thereby the mixing of the fluid is enhanced locally and hence the heat transfer. With an increase in the temperature difference, the peak turbulent stress component in the stream-wise direction shifts downstream of the cylinder and the magnitude of these stresses is also found to decrease. Unlike the flow characteristics, the thermal characteristics show a remarkable difference with an increase in the temperature difference. The vortex shedding process is observed to take place beyond the location x/D = 0.015 due to the presence of a small recirculation region. The diffusion of turbulent heat flux is found to be highest at the edge of the recirculation zone. Also, the velocity fluctuations in the transverse direction are damped significantly just after the end of the recirculation zone. The dissipation rate of transverse velocity fluctuations, and diffusion of the temperature inside the recirculation zone, are found to be smaller compared to that outside. In future, the work will be extended to study the heat transfer from the row of cylinder.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper CRediT authorship contribution statement Sourabh Jogee: Software, Validation, Formal analysis, Investigation, Writing - original draft. B.V.S.S.S. Prasad: Conceptualization, Supervision, Writing - review & editing. Kameswararao Anupindi: Conceptualization, Methodology, Supervision, Writing - review & editing. Acknowledgments The computational resources provided on VIRGO computing cluster by ‘P.G. Senapathy center for computational resources’ at Indian Institute of Technology (IIT) Madras, are gratefully acknowledged. The last author gratefully acknowledges the new-faculty initiation-grant given by IIT Madras (MEE1617852NFIGKAME).

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