1 Introduction

Heat transfer is crucial in many engineering applications like electronic chips (power chips, heat sinks, processors etc.), cooling systems, air conditioning, and heat exchanger systems. As such, heat transfer and flow analysis over isothermal square and circular cylinders got attention because of improving convective heat transfer [1, 2]. Different methods are used to control flow and enhance heat transfer from cylinders [3], and one such type is to alter the geometry of the cylinder or its orientation to vary the attack angle (α). The angle of attack is the angle between the direction of the incoming fluid flow and the surface of the cylinder. Heat transfer characteristics of a square cylinder are, to some extent, comparable to circular cylinder, in particular, at the front surfaces of the cylinders [4]. Because of circular cylinder’s rotational symmetry, the attack angle does not have any effect on the heat transfer and flow characteristics. However, sections of square cylinder are extremely responsive to attack angle with α = 0°–90° as these sections have a 4th order rotational symmetry. The corners and sides of a square cylinder have infinite and zero curvatures, respectively, while the curvature of the circular cylinder is finite (reciprocal of radius). Changing the corner radius ratio (r/R) for a square cylinder from 0.0 to 1.0 generates infinite number of cylinders, where the numerator is the rounded corner radius (r) and the denominator is the radius of the circular cylinder (R), which is equal to the half-width of the square cylinder (R = W/2, Fig. 1b). The corner radius ratio r/R = 0.0 and 1.0 thus represent square and circular cross-section, respectively. When a fluid flows over a circular cylinder, it typically forms a series of vortices on the downstream side of the cylinder, which can cause a fluctuating lift force on the cylinder. This phenomenon is known as the von Karman vortex street. On the other hand, flow over a square cylinder (with α = 0°) can produce a more stable lift force, as the vortices are not as strong and do not fluctuate as much. Additionally, the flow over a square cylinder tends to separate at the corners, which can cause a significant increase in drag. Ozturk, et al. [5] experimentally investigated the flow past a confined cylinder with r/R = 1.0, in a narrow rectangular duct at Re = 1500–6150, using the PIV technique. Ozgoren, M. [6] experimentally studied the flow structure of the near wake region arising from flow past a circular cylinder, a square cylinder, and oriented square cylinder for Re = 550–3400. Due to the functional importance of circular and square cylinders, it is crucial examining the heat transfer and flow characteristics over circular, square, and round-edge cylinders by varying α and r/R [7].

Fig. 1
figure 1

a Schematic of the computational domain. b Cylinder with attack angle α°(deg) and corner radius ratio (r/R). c Typical grid structure in the computational domain

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Zdravkovich et al. [8] classified the laminar flow into four classical flow regimes: creeping laminar flow (0 ≤ Re ≤ 4), laminar flow with steady separation (4 < Re ≤ 48), two-dimensional (2D) periodic laminar vortex shedding regime (48 < Re ≤ 180), and flow with three-dimensional (3D) instabilities (180 < Re ≤ 400). Alam [9] made a compendium and specified the flow regime precisely. He, for a circular cylinder, found non-separation flow at Re ≤ 5, steady wake at 5 < Re ≤ 46.75, two-dimensional periodic laminar vortex shedding at 46.75 < Re ≤ 180, two- to three-dimensional flow transitions at 180 < Re ≤ 225 and 3D flow at Re > 225. Similarly, for a square cylinder, non-separation flow occurs at Re ≤ 1, steady wake at 1 < Re ≤ 49.75, two-dimensional periodic laminar vortex shedding for 49.75 < Re ≤ 165.7, two- to three-dimensional flow transitions at 165.7 < Re ≤ 210 and three-dimensional flow at Re > 210. Zafar and Alam [10] numerically investigated flow over and heat transfer for a cylinder heated at constant temperature placed in tandem to another smaller cylinder, for a spacing ratio 5.5 at Re = 200. Increasing the shear layer velocity and reducing the wake recirculation size leads to enhanced heat transfer.

Dalton and Zheng [11] presented numerical findings for how flow behaves around the diamond and square cylinders at corner radius r/R = 0.25 with Re = 250 and 1000. The corner radius ratio had a larger effect on the wake of the diamond cylinder than that of the square cylinder. Miran and Sohn [12] explored forced oscillations of a cylinder with r/R = 0.0–1.0 and Re = 500 with maximum amplitude of vibration equal to 0.25D. They discovered that in the case of a vibrating cylinder, the Strouhal number (St) increases with increasing r/R and is roughly equal to that of a stationary cylinder. However, the time-mean drag coefficient (C̄D) and fluctuating lift (CL′) for the vibrating cylinder with r/R = 0 are 10% and 40% higher than stationary cylinder’s respective parameters. In all oscillating and stationary cylinder cases, CL′ and C̄D were nominal at r/R = 0.4–0.6. The time-mean drag coefficient of the diamond cylinder was 5.6% lower than that of the square cylinder. In another study, Miran and Sohn [13] explored the influence of r/R on the aerodynamics of a cylinder with square cross-section at Re = 500 and α = 0°–45°. The time-averaged lift (C̄L) and time-mean drag coefficient (C̄D) for r/R = 0 were minimum at a critical attack angle of 12°. With rising r/R, the critical α falls, ranging between 5 and 10° for r/R = 0.2–0.8. Heat transfer from a circular cylinder subjected to a turbulent airflow was investigated in a wind tunnel at Re = 60–25,800 where the results showed how the Nusselt number in turbulent flow and smooth flow varies with Re [14, 15]. Badr [16] numerically studied the effect of freestream turbulence on laminar forced convection from an isothermal tube at Re = 50–500. He found that the surface- and- time mean Nusselt number \(\langle \overline{Nu }\rangle\) increases with increasing Re and freestream turbulence. Different aspects such as vortex shedding, vibrations, wake structure, St, and \(\langle \overline{Nu }\rangle\) have been investigated for engineering applications [17,18,19]. Alam [9] reviewed the fundamentals of flow and heat transfer around and from a square cylinder with recessed, chamfered, rounded and cut corners. He shed light on how these modified corners affect the heat transfer and flow characteristics.

Investigations on the effect of corner radius ratio and attack angle on the heat transfer and flow structures are scarce. Sohankar et al. [20] explored the effect of angle of attack on square cylinder for Re = 45–200 and α = 0°–45°. They found out that the onset of vortex shedding occurs in the range of 40 < Re < 55 depending on α = 0°–45°. A rise in α contributes to a reduction in Re of the onset of the vortex shedding. Electronic components have rapidly evolved over the last few years, achieving a smaller size and improved arrangement. A basic challenge of onboard chips is to prevent overheating and to elongate their working life. The orientation and shape of the components are two factors, which determine heat transfer and flow. In the engineering fields, square cylinders with rounded corners, subjected to a flow with α, have been used. It is thus crucial to study how r/R and α influence fluid dynamics including heat transfer, fluid forces, St, flow separation, etc.

In this work, cylinder with rounded corner (r/R) = 0.5 is used with varying angle of attack (0° ≤ α ≤ 45°). The cylinder is subjected to airflow at a Reynolds number (Re) = 180 and Prandtl number (Pr) = 0.7. The remainder of this paper is organized as follows: In Sect. 2, we presented the problem description and the system of incompressible Navier–Stokes equations with a fully implicit finite element discretization. Furthermore, in this study, sensitivity studies of mesh and time step, along with a validation process through a literature review, are presented for the cylinder case at α = 0°. This validation compares C̄D, CʹD, CʹL, \(\langle \overline{Nu }\rangle\), St and the wake bubble length (\({L}_{b}^{*}\)) between the results obtained in this study and those reported in the literature. In Sect. 3, the CFD simulations are conducted to investigate the dependence of C̄D and \(\langle \overline{Nu }\rangle\) on the flow topology, wake structure, and heat transfer. Some concluding remarks are given in Sect. 4.

2 Numerical methods

2.1 Physical model

Figure 1b shows a 2D cylinder with the length of each side and curved corner radius ratio (r/R) = 0.5. The angle of attack (α) is changed from 0 to 45 \(^\circ\) (Fig. 1b). With increasing α, as the lateral width (D) of the cylinder increases from D/W = 1.0 to 1.21, the inlet velocity (U∞) is decreased accordingly to keep Re = 180 unchanged for different α values. Figure 1a shows a schematic of the flow domain. Structural meshing is done around the cylinder and presented in Fig. 1c. Table 1 presents the size of the flow domain. As presented in Table 1 and Fig. 1a, the upstream and the downstream boundaries are Lu = 11.5D and Ld = 38.5D evaluated from the cylinder’s origin, and H = 20D, provides a blockage ratio (D/H) of 5%, [7, 20]. Since the computational domain size is given based on D, not on W, the blockage ratio remains unchanged for all α values [21].

Table 1 Computational domain size
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2.2 Governing equations

The physics of 2D viscous laminar, incompressible flow with dimensionless form and constant properties over a cylinder is dominated by the continuity, momentum, and energy equations.

The continuity equation:

$$\frac{{\partial u^{*} }}{{\partial x^{*} }}\, + \,\frac{{\partial v^{*} }}{{\partial y^{*} }}\, = \,0.$$
(1)

The momentum equations:

$$\frac{{\partial u^{*} }}{{\partial t^{*} }}\, + \,u^{*} { }\frac{{\partial u^{*} }}{{\partial x^{*} }}\, + \,v^{*} \frac{{\partial u^{*} }}{{\partial y^{*} }}\,{ = }\, - \,\frac{{\partial p^{*} }}{{\partial x^{*} }}\, + \,\frac{1}{Re}\left( {\frac{{\partial^{2} u^{*} }}{{\partial x^{*2} }}\, + \,\frac{{\partial^{2} u^{*} }}{{\partial y^{*2} }}} \right),\quad {\text{and}}$$
(2)
$$\frac{{\partial v^{*} }}{{\partial t^{*} }}\, + \,u^{*} \frac{{ \partial v^{*} }}{{\partial x^{*} }}\, + \,v^{*} \frac{{ \partial v^{*} }}{{\partial y^{*} }}\, = \, - \,\frac{{\partial p^{*} }}{{\partial y^{*} }}\, + \,\frac{1}{Re}\left( {\frac{{\partial^{2} v^{*} }}{{\partial x^{*2} }}\, + \,\frac{{\partial^{2} v^{*} }}{{\partial y^{*2} }}} \right).$$
(3)

The energy equation:

$$\frac{{\partial \Theta^{*} }}{{\partial t^{*} }}\, + \,u^{*} \frac{{ \partial \Theta^{*} }}{{\partial x^{*} }}\, + \,v^{*} \frac{{\partial \Theta^{*} }}{{\partial y^{*} }}\, = \,\frac{1}{Re Pr}\left( {\frac{{\partial^{2} \Theta^{*} }}{{\partial x^{{{*}2}} }}\, + \,\frac{{\partial^{2} \Theta^{*} }}{{\partial y^{{{*}2}} }}} \right).$$
(4)

depends on the incoming velocity U∞, temperature T∞, and projection width D, the parameters are made dimensionless as stream velocity u* = \(u/{U}_{\infty }\), cross-stream velocity v* = \(v/{U}_{\infty }\), and pressure p* = \(p/{\rho U}_{\infty }^{2}\), x* = x/D, and y* = y/D, time \({t}^{*}=t {U}_{\infty }/D\) and temperature \({\Theta }^{*}= \left(T-{T}_{\infty }\right)/\left({T}_{w}-{T}_{\infty }\right)\), where \({T}_{w}\) is the cylinder surface temperature and Re \(=\rho {U}_{\infty }D/\mu\) and \(\mathit{Pr}=\frac{{c}_{p}\mu }{k}\), with ρ, μ, cp and k are the fluid density dynamic viscosity, specific heat and thermal conductivity, respectively.

2.3 Boundary conditions, computational details, and flow parameters

The following boundary conditions are implemented:

$${\text{At}}\,{\text{the}}\,{\text{inlet}},\,u^{*} \, = \,1,\,v^{*} \, = \,0,\,\,{\text{and}}\,\Theta^{*} \, = \,0.$$
(5)
$${\text{At}}\,{\text{the}}\,{\text{outlet}},\,\frac{{\partial \varphi^{*} }}{{\partial x^{*} }}\, = \,0,\,{\text{where}}\,\varphi^{*} \, = \,u^{*} ,\,v^{*} \,and\,\Theta^{*} .$$
(6)

Cylinder surface is prescribed with no slip condition:

$$u^{*} \, = \,0\,{\text{and}}\,v^{*} \, = \,0.$$
(7)

At the upper and lower boundaries:

$$\frac{{\partial u^{*} }}{{\partial y^{*} }}\, = \,0,\,v^{*} \, = \,0,\,{\text{and}}\,\frac{{\partial \Theta^{*} }}{{\partial y^{*} }}\, = \,0.$$
(8)

The surface of the cylinder is assumed to be isothermal, i.e., Θ* = 1.

The coefficients drag and lift (CD and CL) are calculated as CD = 2FD/ρ \({U}_{\infty }^{2}\) D and CL = 2FL/ρ \({U}_{\infty }^{2}\) D. The lift and drag forces FL and FD are determined by integrating the viscous shear force and pressure distributions. The vortex shedding frequency is normalized as Strouhal number St (= fD/U∞), where f is the frequency of vortex shedding the fluctuating lift force. The local heat transfer coefficient (hθ) over a cylinder surface at azimuthal position θ is calculated from:

$$- \,k\,\frac{\partial T}{{\partial n}}|_{w} \, = \,h_{\theta } \left( {T_{w} \, - \,T_{\infty } } \right),$$
(9)

where n refers to the surface normal to the cylinder. The local Nusselt number is obtained as Nuθ = hθ D/k, time-averaged Nusselt number is obtained as \(\overline{Nu} \, = \,\frac{{1}}{\tau }\mathop \smallint \limits_{{0}}^{\tau } {\text{Nu}}_{\theta } {\text{ dt}}\) , surface-averaged Nusselt number as 〈Nu〉 = \(\frac{1}{s}{\int }_{0}^{s}{\text{Nu}}\theta\text{ ds}\), surface- and time-averaged Nusselt number as \(\langle \overline{Nu }\rangle\) = \(\frac{1}{\tau }{\int }_{0}^{\tau }\langle Nu\rangle dt\), where \(s\) is the cylinder surface area, and τ is the integration of time duration that is varied from 60 to 70 vortex shedding cycles after the computations are converged. Finite volume method with structured mesh is used to solve the governing momentum and energy Eqs. (1–4) with the boundary conditions (5–9). The numerical simulations are done using the commercial code, ANSYS-FLUENT (Fluent, 2019). For pressure–velocity coupling SIMPLE algorithm is used. The convective terms in the momentum and energy equations are discretized using the second order upwind method. The second order backward Euler scheme is used for temporal discretization for the transient solution. The convergence criteria for the continuity, x- and y-momentum, and energy equations were restricted below 10–6. The flow was initialized with u* = 1, v* = 0 and Θ* = 0.

2.4 Mesh and validation

A rectangular grid system and O-xy grid system are provided around the cylinder (Fig. 1c), with 320 points evenly set around the cylinder and 120 points radially. The first layer thickness is 0.001D, and the grid size is enlarged radially with an expansion factor of 1.03.

Mesh resolution is decided based on three mesh systems (Mesh 1, Mesh 2, and Mesh 3) at constant time step (∆t) = 0.017 as shown in the first three rows of Table 2. The aerodynamic parameters C̄D, C̄L, CʹD, CʹL, St, and \(\langle \overline{Nu }\rangle\) do not differ between Mesh 2 and Mesh 3. The dependence of the parameters on ∆t (= 0.015, 0.017, and 0.020) with Mesh 2 is obtained in the last 3 rows of Table 2. The values of C̄D, CʹD, CʹL, \(\langle \overline{Nu }\rangle\), and St do not change appreciably between ∆t = 0.017 and 0.020 or between ∆t = 0.015 and 0.017. The time step = 0.017 is adopted here while the results concur well with Alam et al. [7].

Table 2 Mesh and time-step independence test results at Re = 180, r/R = 0.5, and α = 0°
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Table 3 compares C̄D, CʹD, CʹL, \(\langle \overline{Nu }\rangle\), St and the wake bubble length \({L}_{b}^{*}\) between the present study and the literature. The results of the present study are in good agreement with the results available in the literature, with the maximum deviation being 2.404% in \(\langle \overline{Nu }\rangle\) with Abdelhamid et al. [4].

Table 3 Comparison of the present results with those in the literature at Re = 180, r/R = 0.5 and α = 0°
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3 Results and discussions

3.1 Time-mean and fluctuating forces

The time-mean drag coefficient (C̄D) is highly sensitive to α as shown in Fig. 2a. The time-mean drag coefficient decreases from 1.29 to 1.24 at 0° ≤ α ≤ 5° before increasing from 1.24 to 1.5 at 5° ≤ α ≤ 45°. Alam et al. [7] observed that C̄D declines with increasing α for α = 0°–5°, and then increases for α > 5°, with minimum C̄D at α = 5°. The time-mean lift coefficient (C̄L) is negative for 0° < α < 19°, being minimum (C̄L =− 0.02) at α = 12°. The time-mean lift coefficient rapidly increases with increasing α (= 12°–23°) and reaches its maximum (= 0.005) at α = 23°. The variations of CʹD and CʹL are presented in Fig. 2c, d. The fluctuating lift coefficient (CʹD) firstly decreases to a minimum at α = 3° and then increases to a maximum value at α = 20° before declining. The fluctuating lift coefficient diminishes for α < 5° and then grows with α > 5°, following the trend of C̄D (Fig. 2a, d).

Fig. 2
figure 2

Dependence on α° of a time-mean drag coefficient \({\overline{C} }_{D}\), b time-mean lift coefficient \({\overline{C} }_{L}\), c fluctuating drag coefficient \({C{^{\prime}}}_{D}\) and d fluctuating lift coefficient \({C{^{\prime}}}_{L}\)

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3.2 Strouhal and Nusselt numbers

Figure 3 depicts the dependence of St and 〈\(\overline{N} u\) on α. The Strouhal number declines for α = 0°–5° before a rapid increase from 0.178 to 0.188 at α = 5°–10° and a mild decrease from 0.188 to 0.175 at α = 10°–37°. The Strouhal number is globally maximum at α = 10° and minimum at α = 37°. On the other hand, having a small variation at α = 0°–7°, \(\langle \overline{Nu }\rangle\) increases with α for α = 7°–45°. The growth of both St and \(\langle \overline{Nu }\rangle\) occurs largely at α = 7°–12°. The maximum \(\langle \overline{Nu }\rangle\) occurs at the largest α = 45°.

Fig. 3
figure 3

Dependence on α° of a Strouhal Number St and b surface- and time-averaged Nusselt number \(\langle \overline{Nu }\rangle\)

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3.3 Flow structure

The time-averaged streamlines for α (0° ≤ α ≤ 45°) are created to show the variations of the flow structure and separation of the flow over the cylinder surfaces. The variation of α leads to a considerable shift in the flow separation and length of the wake bubble, which aids in classifying the flow structure. Two major flow patterns are discovered when α is increased from 0 to 45° as shown in Figs. 4 and 5. The flow in pattern A separates at the two trailing corners when 0° ≤ α < 8.5° (Figs. 4b, b1; 5a–c) while the flow separation in pattern B occurs from one leading (upper corner) and one trailing (lower corner) corners when 8.5° < α ≤ 45° (Figs. 4c, c1; 5d–h). The border (α = 8.5°) between patterns A and B is identified at the midpoint between two investigated points α = 7° and 10° (Figs. 4a and 5). Flow patterns A and B each consists of two regions on the surface: region I is the boundary layer region and region II is defined as the primary wake bubble region (Fig. 4b1, c1). The dependence of \({L}_{b}^{*}\) on α in Fig. 5 will be discussed later.

Fig. 4
figure 4

a Flow pattern map with changing α°. Time-mean streamlines around the cylinder for b flow pattern A at α = 0°–10°, c flow pattern B at α = 10°–45°

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Fig. 5
figure 5

Time-mean streamlines and the mean x velocity contour around the cylinder for α (= 0°–45°) (a–h), respectively

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3.4 Effects of α on cylinder-surface flow topology

The effect of α on the time-averaged pressure coefficient C̄p, fluctuating pressure coefficient Cʹp, time-averaged skin friction coefficient C̄f, and \(\overline{Nu }\) is shown in Figs. 6 and 7. With increasing α from 0 to 15°, C̄p decreases in the base region (region II; θ = 100°–225°) but increases in the lower side surface (θ = 235°–280°). On the other hand, C̄p grows around the upper leading corner (θ = 45°–90°) and declines around the lower leading corner (θ = 280°–325°). It is observed that the primary wake bubble region II marked at the top of the figure slightly expands with increasing α while the boundary layer region I shrinks (Fig. 6).

Fig. 6
figure 6

Dependence on θ of a time-averaged pressure coefficient\({\overline{C} }_{p}\), b fluctuating pressure coefficient \({C}_{p}^{^{\prime}}\), c time- averaged Nusselt number \(\overline{Nu }\), and d time-averaged skin friction coefficient \(cf\) around the cylinder.at α = 0°–15°. The upper bars depict the zones of regions I and II as a projection for each angle of attack α

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Fig. 7
figure 7

Dependence on θ° of a time-averaged pressure coefficient \(\overline{c }\) p, b fluctuating pressure coefficient CʹP, c time- averaged Nusselt number \(\overline{Nu }\), and d time-averaged skin friction coefficient \({\overline{C} }_{f}\) around the cylinder.at α = 20°–45°. The upper bars depict the zones of regions I and II as a projection for each angle of attack α

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When α is increased from 0 to 5°, the time-averaged pressure coefficient (C̄p) decreases on the side surfaces and trailing corners (θ = 70°–170°; 200°–290°), see Fig. 6b. The scenario is opposite when α is increased from 5 to 15° where C̄p increases on all surfaces (θ = 0°–360°). Figure 6c presents the variation in \(\overline{Nu }\) on the cylinder surface for α = 0°–15°. The surface- and time-averaged Nusselt number (\(\overline{Nu }\)) at θ = 50°–85° and 115°–145° decreases with increasing α from 0 to 15°, while that at θ = 180°, 225°–260°, and 295°–360° increases. A decrease in C̄f with α = 0°–15° appears at θ = 115°–145°. It is observed that C̄f is independent of α (= 0°–15°) at θ = 155°–225°. The time-averaged skin friction coefficient (C̄f) increases with increasing α = 0°–15° at θ = 225°–245°. The time-averaged skin friction coefficient (C̄f) peaks occurring at side surfaces shift to the right (clockwise direction) following the increase of α. The time-averaged Nusselt number (\(\overline{Nu }\)) and time-averaged skin friction coefficient (C̄f) both are sensitive to α (Fig. 6c, d). The surface- and time-averaged Nusselt number 〈\(\overline{N} u\) enhances when C̄f is large. The time- averaged Nusselt number \(\overline{N} u\) and C̄f are always higher at region I as compared to region II (Fig. 6c, d).

At θ = 140°–220°, C̄p decreases with increasing α for α = 20°–37° while not changing with α for α ≥ 37° (Fig. 7a). At θ = 260°–310°, C̄p increases with increasing α. The primary wake bubble region II slightly shifts towards the cylinder base with increasing α. At the cylinder base, C̄p decreases with increasing α while increasing at θ > 230°–270°.

Figure 7a further shows that the time-averaged pressure coefficient (C̄p) at θ = 20°–75° and 180° decreases when α is increased from 20 to 45° while that at θ = 255°–300° increases with α. The time-averaged pressure coefficient C̄p valleys at θ = 90°–115° (upper trailing corner) and θ = 230°–250° (lower trailing corner) shift right with increasing α (= 20°–45°). The primary wake bubble region II does not change in size (θ range) but shifts right at a large α. With increasing α, \(\overline{Nu }\) increases at θ = 0°–70° and 115°–135° but decreases at θ = 270°–315° (Fig. 7c). When α is increased, \(\overline{Nu }\) peak at θ = 90°–115° shrinks and shifts towards the base region, while that at θ = 230°–255° and 295°–360° heightens. Figure 7d presents time-averaged skin friction coefficient (C̄f) distributions on the cylinder surface for α = 20°–45°. The time-averaged skin friction coefficient (C̄f) increases with α at θ = 10°–55° while decreasing at θ = 285°–320°. The maximum variation of C̄f appears at θ = 85°–120° for all α = 20°–45°.

3.5 Wake structure

The vortex shedding, wake recirculation bubble, flow separation, and vortex formation length are important parameters which all impact the flow characteristics, heat transfer, and induced forces on the cylinder. Figure 8 presents the dimensionless instantaneous vorticity \({\omega }_{z}^{*}\) (\(= {\omega }_{z} \times D/{U}_{\infty }\)) contours at the instant of maximum lift force for 0° ≤ α ≤ 45°. The wake is unsteady, characterized by alternating Kármán vortices. In Fig. 8a–j, the minimum or maximum \({\omega }_{z}^{*}\) (\({\omega }_{z \mathrm{min}}^{*}\) or \({\omega }_{z \mathrm{max}}^{*}\)) for the upper-row (negative) and lower-row (positive) vortices are marked to emphasize the vorticity dissipation. On the other hand, in Fig. 8k, l, the magnitudes of \({\omega }_{z}^{*}\) of the first (negative) and second (positive) vortices are plotted against α. For α = 0°–5°, |-\({\omega }_{z}^{*}\)|max rapidly decreases from 2.40 to 2.32 and then rapidly increases from α = 5° to 7° (Fig. 8a–d, k). For α = 0°, the first two shed vortices (one negative and one positive) have |-\({\omega }_{z}^{*}\)|max = 2.40 and \({\omega }_{z \mathrm{max}}^{*}\)= 1.89, respectively, indicating a difference between the two sides (Fig. 8a, k).

Fig. 8
figure 8

Dependence on α of the isocontours of normalized instantaneous vorticity \({\omega }_{z}^{*} = {\omega }_{z} \times d/{U}_{\infty }\), for a–j α = 0°–45°. k Variation in maximum vorticity (negative) with α°. l Variation on maximum vorticity (positive) with α. m Dependence of streamwise length \({L}_{b}^{*}\) of recirculation bubble on α°

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One global minimum of |-\({\omega }_{z}^{*}\)|max appears at α = 5°, two local minimums occur at α = 10° and 28°, and two peaks at α = 7° and α = 17°, respectively. For α = 10°, |-\({\omega }_{z}^{*}\)|max and \({\omega }_{z \mathrm{max}}^{*}\) are 2.37 and 1.99, respectively (Fig. 8e, k). It suggests that a difference in vorticity dissipation between the two sides is smaller at α = 10° than at α = 0°. A similar observation is made for α = 15° (Fig. 8f, k). For α = 30°–45°, |-\({\omega }_{z}^{*}\)|max grows to 2.40, close to that at α = 0° while \({\omega }_{z \mathrm{max}}^{*}\) remains constant at 2.07 (Fig. 8j, k). The maximum normalized instantaneous vorticity \({\omega }_{z \mathrm{max}}^{*}\) is independent of α at α = 0°–5° while rapidly increasing for α = 5°–30. Interestingly, |-\({\omega }_{z}^{*}\)|max is in an inverse relation to the streamwise length \({L}_{b}^{*}\) of the recirculation bubble (Figs. 5, 8k, m).

Contour plots of fluctuating (root-mean-square) streamwise velocity \(\overline{{u^{\prime}*}}\) for different α values are shown in Fig. 9a–i. Each \(\overline{{u^{\prime}*}}\) field displays two \(\overline{{u^{\prime}*}}\) peaks corresponding to the time-mean rolling positions of the shear layers from the two sides of the cylinder [19, 22,23,24]. The shear layer rolling may play a role in heat transfer from the cylinder base surface (regime II); the nearer the rolling, the higher \(\langle \overline{Nu }\rangle\). The peak value of \(\overline{{u^{\prime}*}}\) represents the strength of rolling, hence has a significant influence on \(\langle \overline{Nu }\rangle\). The peak \(\overline{{u^{\prime}*}}\) values are plotted in Fig. 10a as a function of α. For Pattern A (0° ≤ α < 8.5°), \(\overline{{u^{\prime}*}}\) is almost constant as is \(\langle \overline{Nu }\rangle\) (Figs. 3b, 9a, b, c, 10a). For pattern B, peak \(\overline{{u^{\prime}*}}\) rapidly increases for 8.5° ≤ α ≤ 37° but mildly for 37° < α ≤ 45°, similarly to the trend of \(\langle \overline{Nu }\rangle\) (Figs. 3b, 9d, 10a). The wake size could be defined by vortex formation length and wake width [22, 25]. The vortex formation length \({L}_{f}^{*}\) (= Lf /D) shows the streamwise distance between the cylinder center and uʹ* peak (Fig. 9a). The lateral distance between two \(\overline{{u^{\prime}*}}\) peaks define the wake width w* (= w/D). The vortex formation length is inversely linked with St [4]. With increasing α, \({L}_{f}^{*}\) firstly increases and reaches its maximum at α = 3° before being minimum at α = 15° (Fig. 10b). A slight increase in \({L}_{f}^{*}\) is observed for α = 15°–45°. With increasing α, the wake width w* increases dramatically in pattern B but not much in pattern A (Fig. 10c).

Fig. 9
figure 9

Contour plots of fluctuating streamwise velocity \(\overline{{u^{\prime *} }}\) for α (= 0°–45°) (a–j), respectively

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Fig. 10
figure 10

a Peak \(\overline{{u^{\prime } }}^{*}\) variation with α°. b Vortex formation length \({L}_{f}^{*}\) variation with α°. c Wake width w* variation with α°

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3.6 Momentum and heat transport

The effect of α on the momentum and heat transport topology is presented here. As mentioned before, the wake is unsteady due to the vortex shedding. The momentum and heat are transported through the vortex shedding. The velocity and temperature fields can be decomposed as

$$u\left( t \right)\, = \,\overline{u}\, + \,u^{\prime},$$
(10)
$$v\left( t \right)\, = \,\overline{v}\, + \,v^{\prime}\quad {\text{and}}$$
(11)
$$\Theta \left( t \right)\, = \,\overline{\Theta }\, + \,\Theta^{\prime},$$
(12)

where (\(u\left(t\right)\),\(v\left(t\right)\),\(\Theta \left(t\right)\)) are instantaneous components and (\(\overline{u }\),\(\overline{v }\),\(\overline{\Theta }\)) are the time-mean components, and (\({u}^{^{\prime}}\),\({v}^{^{\prime}}\),\({\Theta }^{^{\prime}}\)) are the fluctuating components. Various Reynolds averaged quantities are measured, including Reynolds normal stresses (\(\overline{{{\text{u}}^{\prime}*{\text{u}}^{\prime}*}}\),\(\overline{{{\text{v}}^{\prime}*{\text{v}}^{\prime}*}}\)) and shear stresses (\(\overline{{{\text{u}}^{\prime}*{\text{v}}^{\prime}*}}\)).

Figure 11 shows the dependence of α on \(\overline{{{\text{u}}^{\prime}*{\text{u}}^{\prime}*}}\). Maximum \(\overline{{{\text{u}}^{\prime}*{\text{u}}^{\prime}*}}\) are marked on the displayed contours with maximum peak value of \(\overline{{{\text{u}}^{\prime}*{\text{u}}^{\prime}*}}\) is 0.264 at α = 45° (Fig. 11). The value of \(\overline{{{\text{u}}^{\prime}*{\text{u}}^{\prime}*}}\) is simply square of \(\overline{{u^{\prime}*}}\) as shown in (Figs. 10a and 11j). The Normalized Reynolds normal stresses \(\overline{{{\text{u}}^{\prime}*{\text{u}}^{\prime}*}}\) is constant in pattern A (0° ≤ α < 8.5°) then increases in pattern B.

Fig.11
figure 11

Normalized Reynolds normal stresses \(\overline{{u^{\prime *} u^{\prime *} }}\) for α (= 0°–45°) (a–j), respectively. j Peak \(\overline{{\text{u}}^{\prime}\text{*}{\text{u}}^{\prime}\text{*}}\) variation with α °

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Figure 12 shows \(\overline{{\text{v}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}}\) contours for different α values. The Normalized Reynolds normal stresses \(\overline{{\text{v}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}}\) largely occurs around the wake centerline with maximum peak value of \(\overline{{\text{v}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}}\) is 0.486 at α = 45° (Fig. 12a–j), while they are plotted in Fig. 12k. The maximum \(\overline{{\text{v}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}}\) first decreases for α < 5° and then increases with α for α > 5°, behaving similarly to C̄D, CʹL, and \(\overline{{u^{\prime}*}}\) (Figs. 2 and 12k). Figure 13 shows variations of \(\overline{{\text{u}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}}\) with α. The negative and positive stresses are shown by dashed and solid lines, respectively. The Normalized Reynolds shear stresses \(\overline{{\text{u}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}}\) gives a degree of correlation between streamwise and cross-stream fluctuating velocities. As a result of the alternate vortex shedding, two peaks emerge in each \(\overline{{\text{u}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}}\) field. The absolute values of \({\left|\overline{{\text{u}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}} \, \right|}_{\mathrm{max}}\) increases with increasing α and is independent of α at 3° < α < 10° and 25° < α < 45° but rapidly increases for α < 3° and 10° < α < 25° (Fig. 13k). The absolute values of \({\left|\overline{{\text{u}}^{\prime}\text{*}{\text{v}}^{\prime}\text{*}} \, \right|}_{\mathrm{max}}\) varies from 0.06 to 0.012 as α increases from 0 to 45°.

Fig.12
figure 12

Normalized Reynolds normal stresses \(\overline{{v^{\prime *} v^{\prime *} }}\) for α (= 0°–45°) (a–j), respectively. k Peak \(\overline{\text{v}^{\prime}{*}\text{v}^{\prime}{*}}\) value variation with α°

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Fig. 13
figure 13

Normalized Reynolds shear stresses \(\overline{{u^{\prime *} v^{\prime *} }}\) for α (= 0°–45°) (a–j), respectively. k peak \(\left|\overline{\text{u}^{\prime}{*}\text{v}^{\prime}{*}} \, \right|\mathrm{max}\) variation with α°

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Figure 14 presents the dependence of streamwise heat flux (\(\overline{\text{u}^{\prime}*\Theta^{\prime} {*}}\)) on α. There are two peaks in each field, with the negative peak magnitude being smaller than the positive peak. The Normalized time-mean streamwise heat fluxes \({\overline{{u }^{{^{\prime}}*}{\Theta }^{{^{\prime}}*}}}_{max}\) is minimum at small α (= 0°–4°) and maximum for α = 30°—45°. The Normalized time-mean streamwise heat fluxes grow with increasing α up to α = 30°. For α ≥ 30°, \({\overline{{u }^{{^{\prime}}*}{\Theta }^{{^{\prime}}*}}}_{max}\) is independent of α (Fig. 14k). The variation of \({\overline{{u }^{{^{\prime}}*}{\Theta }^{{^{\prime}}*}}}_{max}\) is similar to that of \({\overline{{u }^{{^{\prime}}*}{v}^{{^{\prime}}*}}}_{max}\), ensuring that there is a relation between the momentum and heat transport (Figs. 13 and 14). Heat transfer is aided by the vortex shedding.

Fig.14
figure 14

Normalized time-mean streamwise heat fluxes \(\overline{{u^{\prime *} \Theta^{\prime *} }}\) for α (= 0°–45°) (a–j), respectively. k dependence of \(\overline{{u^{\prime *} \Theta^{\prime *} }}_{\max }\) peak on α °

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The contours of cross-stream heat fluxes \(\overline{{\text{v}}^{\prime}{* }\Theta^{\prime} {*}}\) are shown in Fig. 15. The magnitudes of \(\overline{\text{v}^{\prime}*\Theta^{\prime}{*}}\) within the recirculation bubble are weak, with two feeble peaks, as predicted. The peaks outside of the recirculation bubble are very high, showing that velocity and temperature have a very strong relationship outside of the recirculation bubble. As such, \(\overline{\text{v}^{\prime}*\Theta^{\prime}*}\) >\(\overline{\text{u}^{\prime}*\Theta^{\prime} {*}}\) exhibiting that the heat is transported effectively in the lateral direction as compared to the longitudinal direction. The value of \({\left|\overline{\text{v}^{\prime}*\Theta^{\prime}{*}}\right|}_{\mathrm{max}}\) increases with increasing α and is minimum (= 0.02) at small α = 0°–5° and is maximum about 0.035 at largest α = 45° (Fig. 15k).

Fig.15
figure 15

Normalized time-mean cross-stream heat fluxes \(\overline{{v^{\prime *} \Theta^{\prime *} }}\) for different α (= 0°–45°) (a–j), respectively. k peak \(\left|\overline{{V }^{{^{\prime}}*}{\Theta }^{{^{\prime}}*}}\right|\mathrm{max}\) value variation with α°

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4 Concluding remarks

In this work, the flow around and heat transfer from an isothermal rounded cylinder with corner radius ratio 0.5, are investigated with the angle of attack varying from 0 to 45°. The time-mean drag coefficient decreases from 1.29 to 1.24 at 0° ≤ α ≤ 5°, after that increases from 1.24 to 1.50 at 5° ≤ α ≤ 45°. The time-mean lift coefficient becomes minimum at α = 12° and maximum at α = 22°. The fluctuating drag force decreases and reaches its minimum at α = 3° before increasing to its maximum at α = 20°. The fluctuating lift force first declines and becomes minimum at α = 5°, followed by a rapid increase upto α = 37°.

The Strouhal number drops with rising the angle of attack for α = 0°–5° and then increases to its maximum at α = 12°. The surface-averaged Nusselt number is weakly dependent on α for α = 0°–7°, while it rapidly increases with increasing α for α = 7°–45°. The flow structure can be divided into two major flow patterns A and B. Flow pattern A occurs at 0° ≤ α < 8.5°, termed as the “trailing corner separated flow”, while pattern B appears at 8.5° < α ≤ 45°, termed as “leading-trailing corner separated flow”. The flow is unsteady, characterized by alternating Kármán vortices at each α. The maximum vorticity is inversely linked to the streamwise size of the recirculation bubble.

The variations of the maximum fluctuating streamwise normalized Reynolds normal stress in x-direction and the normalized Reynolds normal stress in y-direction are highly correlated with heat transfer. The maximum fluctuating streamwise and normalized Reynolds normal stress in x direction are constant for flow pattern A, while they increase with increasing α for pattern B. The peak of the normalized Reynolds shear stresses, the peak of the normalized time-mean streamwise heat flux in x direction, and the peak of the normalized time-mean streamwise heat flux in y direction, increase with increasing α upto α = 25°, 30°, and 35°, respectively. This displays a connection between the momentum and heat transport over the cylinder surface. The vortex formation length reaches its maximum at α=3° and minimum at α = 17°, and independent of α for α = 20°–37°. As future work, we plan to extend this study to investigate the flow structure and heat transfer of two- or three-square cylinders in various positions, with different angles of attack and corner radius ratios at low Reynolds numbers.