Influence of turbulence intensity and turbulence length scale on the drag, lift and heat transfer of a circular cylinder

Influence of the turbulence intensity and turbulence length scale on the hydrodynamic characteristics and heat transfer of a circular cylinder, streamlined by a viscous fluid flow, is numerically studied. We take the Reynolds number of the oncoming flow equal to 4×104, the turbulence intensity Tuf and the dimensionless turbulence length scale L̅f varying from 1.0% to 40% and from 0.25 to 4.0, respectively. We confirmed that the increase of Tuf leads to the suppression of the periodic vortex shedding from the cylinder surface, and as a result the stationary mode of streamlining is formed. Consequently, with the increasing turbulence intensity directly in front of the cylinder Tu*, the amplitude of the lift coefficient decreases monotonically. Nevertheless, the time-averaged drag coefficient of the streamlined cylinder decreases at Tu*<6.0%, and increases at Tu*>9.0%. The dependence of the average Nusselt number on Tu* is near-linear, and with the increasing turbulence intensity, the Nusselt number rises. However, the change of the average Nusselt number depending on L̅f is non-monotonic and at L̅f, the value reaches its maximum.


Introduction
Hydrodynamics of flows and their heat transfer with bodies are widely present in nature, technology, various thermal engineering facilities, heat exchangers, hot-wire anemometers, and other instruments and apparatus. In many cases, the bodies are cylindrical in shape and the fluid motion is turbulent. The investigation of peculiarities of transverse flow around circular cylinders and their thermal interaction with fluid is of great interest.
In accordance with current ideas, when a turbulent flow streamlines the body (Kutateladze, 1990) as the fluid approaches it, the main flow turbulence becomes distorted, interacts with the body surface, influences the boundary layer and position of its separation point and it also affects the flow in the wake behind the body.
Accordingly, it is possible to define the following typical regions of the flow: the upstream region with the high-Reynolds-number and isotropic flow; the region in the vicinity of the cylinder where the vortices are stretching, and a turbulent anisotropy occurs; the region close to the cylinder surface where the vortices interact with each other and penetrate the boundary layer.
Consequently, the turbulence of the external flow may differ significantly from the turbulence near the surface of the cylinder. The distribution of the near-wall turbulence is determined depending on the processes intensity of the turbulent energy generation, its dissipation in the external flow, the shape of the cylinder and the type of the wall boundary conditions in the boundary layer.
At the present time, there are a lot of researches in this field, but they are mainly experimental. The studies are focused on the influence of the turbulence intensity Tu f and the turbulence length scale L f of the oncoming flow on the hydrodynamic characteristics, and the heat transfer of transverse flow around circular cylinders. Ko and Graf (1972) and Yeboah et al. (1996) showed that for relatively low Reynolds numbers, the increase of Tu f causes the increase of the cylinder drag coefficient C D . And on the contrary, at high Reynolds numbers, the cylinder drag coefficient decreases with the increasing Tu f (Surry, 1972;Younis, 2010).
Regarding the impact of the turbulence length scale on the drag coefficient, Ko and Graf (1972) stated that the effect of , where D is the streamlined cylinder diameter, is expressed to a lesser extent as compared with Tu f . These scientists have found that for the above-mentioned Reynolds numbers, the lowest drag coefficient of the circular cylinder is observed in experiments at and Tu f =4%. According to Arie et al. (1981), at low Reynolds numbers, the value of C D reaches the minimum when varies within the range from 0.25 to 0.78. It should be noted here that results of Arie et al. (1981) and Ko and Graf (1972) are similar. Surry (1972) and Savkar et al. (1980) stated that for high Reynolds numbers, the reduction of leads to the decrease of the cylinder drag coefficient. Younis (2010) specified that at sufficiently high turbulence intensity (Tu f =9%), the cylinder drag coefficient can decrease by approximately 74.7% due to the reduction of to 0.35 compared with the case of non-turbulized flow. If the turbulence intensity is low (Tu f =5%), then with the increasing , the drag coefficient increases and reaches the maximum value that exceeds the value of C D of a cylinder for a nonturbulized flow by 18.4%.
As it can be seen from these results, the drag coefficient of a streamlined circular cylinder depends on the Reynolds number, the turbulence intensity and the turbulence length scale in a complicated manner.
It is known (Zhukauskas, 1982 and order) that one of the main directions of heat transfer enhancement on the surface of a cylinder is the turbulization and swirling of the oncoming flow. The heat transfer of a heated cylinder depends on the Reynolds number, the Prandtl number, the turbulence intensity, the turbulence length scale, and other factors.
Until recently, the heat exchange between the heated cylinders and the flow has been studied mostly experimentally, and the amount of the numerical studies is relatively small. In the experiments, various lattices, grids, and other devices were used to increase the free-stream turbulence intensity. The flow is considered to be weakly turbulized if Tu f <1% and when Tu f >10% it is strongly turbulized.
During the experiments in the first place, the attention was paid to the behaviour of local heat transfer coefficients depending on the above-mentioned parameters and the turbulence intensity Tu f (Smith and Kuethe, 1966;Kestin and Wood, 1971;Dydan and Epick, 1970;Dydan et al., 1974;Boulos and Pei, 1973) more often from the windward side and at the front point of a cylinder.

Nu
When studying the heat transfer of cylinders, the heat transfer characterized by the average Nusselt number arouses interest.
According to the experimental data obtained by Kondjoyan and Daudin (1995) in the case of a transverse flow past a single circular cylinder at 4×10 3 <Re<3×10 4 , the average Nusselt number is defined by using the following for-mula: where the multiplier C and the exponent n depend on the turbulence intensity; , is the average local convective heat transfer coefficient, is the thermal conductivity coefficient of the fluid, and R is the cylinder radius; Reynolds number , u f is the oncoming flow velocity, is the kinematic viscosity of the fluid. In particular, when Tu f =1.5%, c is equal to 0.62, n is equal to 0.506. For , the linear turbulence length scale is comparable with the diameter of the cylinder D. k − ε Nu Nu From the experimental results of Kondjoyan and Boisson (1997), it follows that at high Reynolds numbers, the increase of free-stream turbulence intensity causes a significant increase of the average Nusselt number, though at low Reynolds numbers, the increase is small. In this paper, the results of numerical experiments carried out by using the turbulence model and the standard wall functions are presented. They show that compared with the data of physical experiments, the numerical values of are much larger, in some cases, the relative difference reaches 225%. It is indicated that the value of depends on the linear turbulence length scale to a lesser degree than on the turbulence intensity.
Moreover, the dependence of the average Nusselt number on the Reynolds number, the turbulence intensity, and the turbulence length scale is given by Sak et al. (2007), Lowery and Vachon (1975), Mehendale et al. (1991), and Sikmanovic et al. (1974). According to Sikmanovic et al. (1974), at , , : where . From the review, it follows that when analyzing the influence of free-stream turbulence on the hydrodynamics and the heat transfer of the transverse flow around a circular cylinder, the main attention is paid to the drag, heat exchange at the front point, and the heat transfer of the cylinder. At the same time, the turbulence intensity distribution in the flow, across the cylinder surface, the influence of the turbulence intensity and the linear turbulence length scale on the lift force and other indices are of great interest. Some of these issues have been examined by Morenko and Fedyaev (2010). Undoubtedly, all of the above can also be applied to the case of flow around vibrating (Xu et al., 2012) and porous cylinders (Zhao et al., 2012).

Governing equations and method of solution
Reynolds-averaged continuity equations are used as the governing equations for the two-dimensional unsteady turbulent incompressible flow around the circular cylinder laid on a plane, which can be written in the following form in the Cartesian coordinate system Turbulent heat transfer is modeled by using the concept of the Reynolds analogy to turbulent momentum transfer ∂T ∂t where t is time; x and y denote the horizontal and vertical directions, respectively; u and v are the velocity components; p is pressure; T is the fluid temperature; is the fluid density; a is the thermal diffusivity; , , and are the turbulent stresses; q x * and q y * are the turbulent heat flow components, and .
In the present computations, the shear stress transport (SST) turbulence model is used (Menter, 1994). The SST turbulence model combines the advantages of both the standard model and the model. To overcome the deficiencies of both models, the SST model uses a blending function for gradual transition from the standard model near the wall to a high Reynolds number version of the model in the outer portion of the boundary layer.
It should be noted that according to the definition of the turbulence intensity of kinetic energy, we have . Ideally, the turbulence intensity should be determined experimentally. Velocity of the turbulent dissipation , kinetic energy k and the dimensionless linear turbulence length scale are related by the equation . Since , the specific turbulent dissipation and the dimensionless linear turbulence length scale are related by . The computational domain is a rectangle with a circular cylinder inside. Its diameter D is assumed to be 0.2 m. The outer boundaries of the computational domain are considered to be at a sufficient distance from the cylinder, therefore their impact on the numerical solution of the problem is negligible. The computational domain is extended to 10D and 40D from the upstream and downstream of the cylinder surface, respectively. The height of the computational domain is 20D.
The following boundary conditions are applied. No-slip boundary condition is set by specifying the velocity components to be zero, i.e. u=0, , at the fluid-wall and fluid-cylinder interfaces. The surface temperature of the cylinder is constant: .
At the inlet boundary, the horizontal velocity components are assumed uniform u=u f and the vertical velocity , the working pressure , the fluid temperature , free-stream turbulence intensity Tu f and the dimensionless linear turbulence length scale . With given val- ues of u f , Tu f , and , the kinetic energy and the specific turbulent dissipation rate are determined at the inlet of the computational domain. The symmetry condition is used at the top and bottom boundaries of the computational domain. At the outlet boundary, the free flow condition is used as a free boundary condition.
The initial condition for all the computations is an impulsive start, at t=0 the velocity is assigned the value that corresponds to the potential flow around a stationary cylinder.
The solution of the problem is based on the method of finite volumes. Irregular triangular meshes with significantly large number of nodes in the vicinity of a cylinder are used when performing the calculations. The total number of the computational meshes' nodes is about 250000. The number of nodes on the cylinder surface is 200.

Calculation results
Velocity and pressure fields, airflow streamlines, vorticity, turbulence intensity, drag and lift coefficients, local and average Nusselt numbers are calculated when airflow transversely flows around the heated circular cylinder for Re=4×10 4 before the critical region of Reynolds numbers are characterized by transition of boundary layer from laminar to turbulent with a sharp fall of the pressure drag. The turbulence intensity and the free-stream linear dimensionless turbulence length scale vary from 1.0% to 40% and from 0.25 to 4.0, respectively. The following air parameters are specified: density =1.225 kg/m 3 , dynamic-viscosity coefficient =1.7894×10 -5 kg/(m·s), thermal conductivity coefficient =0.0242 W/(m·K), and specific heat at constant pressure c p =1006.43 J/(kg·K). The air temperature at the inlet section of the computational domain is assumed to be T f =300 K, and the cylinder surface temperature is T w =350 K. The Prandtl number of the medium practically does not change and, in this case, is equal to 0.74.

L f
The results of preliminary calculations performed at Tu f =1.5% and =1, when the oncoming flow is not strongly turbulized and there is no additional turbulence showed that the flow regime is self-oscillating, and Karman vortex street is formed in the cylinder wake (see Fig. 1). In general, the calculated data are in satisfactory agreement with the results of other authors (Kutateladze, 1990;Braza et al., 2006).
The increase of Tu f from 1.0% to 40% leads to the suppression of the periodic vortex separation from the cylinder surface and the formation of stationary mode flowing around (see Fig. 2). Thus, depending on the turbulence intensity viscous fluid flow around a cylinder can be both self-oscillating and steady.
According to the experimental data of Kondjoyan and Daudin (1995) behind a perforated plate that allows generating a higher level of turbulence in the oncoming flow, the turbulence intensity gradually decreases downstream. This is also confirmed by the results of the numerical calculations. However, the presence of a circular cylinder in the flow leads to the fact that the turbulence intensity changes largely near the cylinder surface (see Figs. 1b, 2b and Fig. 3). In general, apart from the remoteness of a cylinder from a turbulator, the flow history effect, the presence of external boundaries and other factors influence its value. Therefore, when analyzing the effect of the turbulence intensity on the hydrodynamics and the heat transfer of the cylinders, it is more appropriate to estimate it based on the turbulence intensity directly in front of the cylinder Tu * , for example, at a distance of 2R-3R from the front point. However, accord-ing to the results Tu * (0.77 − 0.011Tu f ) Tu f .
3.1 Turbulence intensity change at the cylinder surface The turbulence intensity change at the cylinder surface depending on the circumferential coordinate at different values of Tu * is illustrated in Fig. 4. From the above data it follows that the increase of Tu * promotes the increase of the turbulence intensity at the cylinder surface, mainly on the windward side. At the stagnation point ( =0), a local minimum of the turbulence intensity is observed, and then its value increases to the maximum followed by the decrease of Tu. The position of the minimum of function Tu is close to the separation point of the laminar boundary layer ( 2.2). After that, the turbulence intensity increases, reaching the local maximum at the rear point ( =3.14).
3.2 Effect of turbulence intensity on the time-averaged drag coefficient of a streamlined cylinder Effect of the turbulence intensity Tu * on the time-averaged drag coefficient C D of the streamlined cylinder is illustrated in Fig. 5. There are considerably nonlinear changes of , depending on Tu * , where F x is the projection of a hydrodynamic force of the cylinder. With the increasing turbulence intensity the drag coefficient decreases at Tu * <6.0%, and increases at Tu * >9.0%. Accordingly, within the range of 6.0%<Tu * <9.0%, C D reaches the minimum when the stationary mode of the flow around a cylinder is established.

Effect of the turbulence intensity on the lift coefficient
As noted above, apart from the drag coefficient, the lift coefficient C L is also of interest. It is defined as , where F y is the projection on the y-axis of a hydrodynamic force acting on the cylinder. Fig. 6 shows the behavior of the lift coefficient, depending on time for different values of the turbulence intensity Tu * (Fig. 6a), the amplitude C L * of   the lift coefficient when Tu * changes (Fig. 6b). It is seen that over time the lift coefficient changes in the self-oscillating mode. With the increasing turbulence intensity, the amplitude of the lift coefficient decreases by the law that is similar to the linear one, which means the suppression of the Karman vortex street. When Tu * ≈14%, the amplitude C L * is close to zero and when Tu * =16%, the flow is symmetrical, and the Karman vortex street is absent.

Effect of the turbulence intensity on the heat transfer φ
The Nusselt number Nu along the cylinder surface changes as follows (Fig. 7). A local maximum is observed at the stagnation point of the flow. As the boundary layer thickness increases, Nu decreases. The minimum is reached at =1.6-1.8 in the vicinity of the flow separation point from the surface. Further, Nu increases, reaching the local maximum at the rear point. It can be seen that with the increasing turbulence intensity, the heat transfer increases to a large extent on the windward side of the cylinder.
It is also found that with the increasing Tu * (Fig. 8), the average Nusselt number over the cylinder surface increases, according to the linear law at Tu * >2.5%:

Nu
In general, when Tu * changes from 1.0% to 16%, increases by 17%. If in Eq. (8), using Eq. (7) Tu * is substituted by Tu f , a quadratic dependence of the average Nusselt number on Tu f is obtained, and in this case, the Nusselt number can reach the maximum value. Thus, the results of calculations and experiments show that when the free-stream turbulence intensity changes, at certain values of Tu f , the maximum heat transfer of a streamlined circular cylinder is possible.
The results of comparison of the calculated values of with the experimental data at Re=27700, =0.78 are presented in Fig. 9.
Despite the fact that the approximating dependence (2) was obtained for Re=19000, <0.19, the Nusselt numbers are close to the calculated values of . These Nusselt numbers differ most of all from the values calculated by Kondjoyan and Daudin (1995), by approximately 52.0%. It should be noted that the results obtained in the present study exceed the experimental data which, in general, are consistent with the conclusions of Kondjoyan and Boisson (1997). There is a satisfactory agreement between the calculated and experimental data both in the magnitude order of , and the behavior of the Nusselt number change depending on Tu f . Despite the fact that this parameter is believed to have less influence on the hydrodynamics and heat transfer of the cylinders than the turbulence intensity, it should be taken into consideration in the calculations. Fig. 10 shows the change of the average Nusselt number, depending on the free-stream linear turbulence length scale at Tu f =1.5%.
At first, the heat transfer of the cylinder increases with the increase of to 1.0, mainly because of the heat transfer enhancement from the windward side. At ≈1.0, the value of reaches the maximum, and then it decreases since the heat transfer decreases at the rear area of the cylinder. Sanitjai and Goidstein (2001) and Žukauskas et al. (1993) also noticed the presence of the maximum when changing the linear turbulence length scale and the fixed turbulence intensity. However, the maximum heat transfer in these studies is observed at =1.6, 0.06, respectively. In our I.V. Morenko, V.L. Fedyaev China Ocean Eng., 2017, Vol. 31, No. 3, P. 357-363 opinion, the difference can be explained by different Reynolds numbers and the turbulence intensity.

Conclusions
From the results of computational experiments, it follows that the free-stream turbulence intensity has a significant influence on all main characteristics of the streamlined cylinder. Depending on the value of Tu f , the flow regime can be both self-oscillatory and steady. In fluid flow, the turbulence intensity undergoes significant changes; it rises sharply on the windward side of the cylinder, near its surface. When analyzing the hydrodynamics and the heat transfer of the streamlined cylinder depending on the turbulence intensity to eliminate the influence of the distance from its turbulator, the channel walls, and other factors, it is proposed do not estimate the corresponding indicators with respect to Tu f , but with respect to the turbulence intensity Tu * at the distance of two or three radiuses from the cylinder front point.
L f L f It is found that for a fixed Reynolds number, the linear turbulence length scale, the change of Tu * from 1.0% to 16.0%, the drag coefficient of the cylinder in the range Tu * of 6.0%-9.0% reaches the minimum, and the amplitude of the lift coefficient decreases according to the law that is close to the linear one up to zero. In contrast, the average Nusselt number over the cylinder surface increases with the increasing Tu * , and for Tu * >2.5% according to the linear law. The dependence of the average Nusselt number on the dimensionless turbulence length scale for the fixed Reynolds number and the free-stream turbulence intensity is more complicated. At <1.0 with the increasing turbulence length scale, the Nusselt number increases sharply, reaching the maximum, approximately, at =1.0. Then the Nusselt number and, correspondingly, the heat transfer slightly decrease.
In conclusion, the turbulization of a fluid flow oncoming on the circular cylinder is one of the effective factors to influence the nature and structure of the flow, the hydrodynamic forces acting on it, and the heat transfer between the cylinder and the flow. The turbulization allows to minimize the effect of the hydrodynamic forces on the cylinder and to ensure the maximum heat transfer of the cylinder.