Abstract
The behavior and characteristics of the wake produced behind an elliptic cylinder at 0° angle of attack, in the presence of trip wires, are investigated experimentally and numerically. For this purpose, an aluminum cylinder model with an elliptic cross section and a length of 390 mm, major diameter of 42.4 mm and minor diameter of 21.2 mm was tested in the test section of a wind tunnel. Realizable k–ε turbulence model was used for the numerical analysis and SIMPLEC algorithm was employed for pressure–velocity coupling. Reynolds numbers corresponding to the major diameter of the cylinder were 38,550 and 64,250 for air velocities of 15 m/s and 25 m/s, respectively. Turbulence trip wires of .5 mm diameter were installed symmetrically on the elliptic cylinder at angles of 23.7° and 40.9° relative to the stagnation point. The drag coefficient for the smooth elliptic cylinder was about .6 at both Reynolds numbers. Results showed a drag coefficient reduction of 75% in the best possible case. It was also concluded that a turbulence promoter wire has a significant effect on flow characteristics and reduction in the drag coefficient, and that it strongly depends on the location where the trip wire is installed on the model.
Introduction
Investigation of aerodynamics around circular and elliptic cylinders, with their numerous uses in various industries, is an important and common subject of research. Likewise, reduction in drag coefficient (resistance against air) is sought in many industrial and scientific applications. It can lead to reduced fuel consumption in various industries, better heat transfer, noise reduction, increased velocity and enhancement of structural stability. Another factor that plays a major role in the aerodynamics of cylinders is boundary layer and its separation; and, for achieving higher efficiencies, it has always been attempted to prevent boundary layer separation or to delay it as much as possible. By doing so, the drag coefficient can be reduced and optimized. There are different techniques for preventing or delaying boundary layer separation. They include the use of bladeshape vortex generators, surface roughness, trip wires, air flow ionization, and fluid injection into or suction from surfaces. Elliptic cylinders are employed in various applications, including the tubes used in heat exchangers and leading edges of flaps in multisection airfoils, etc.
A Review of Previous Works
Ota et al. (1987) experimentally investigated the flow around an elliptic cylinder of axis ratio 1:3 in the critical Reynolds (Re_{cr}) regime along the cylinder surface and the near wake. They obtained the proper angle of attack for reaching the critical Re, followed by the shedding of first vortices behind the model and found that at the critical Re, the drag coefficient changes discontinuously. In a numerical study on an elliptic cylinder, Paul et al. (2014) analyzed the effects of angle of attack, aspect ratio and Re on flow characteristics, in a laminar and unsteady flow. They also demonstrated the dependence of flow and vortex shedding parameters on the angle of attack. Using vortex generators, Igarashi (1985) conducted a series of experiments on flows around a circular cylinder at subcritical Re numbers where the drag coefficient was reduced by about 53.8% in the best case. In a numerical study of an elliptic cylinder, Raman et al. (2013) showed that at a specific Re, by reducing the cylinder diameter ratio, the drag coefficient is reduced. The velocity and turbulence in the boundary layer around an elliptic cylinder having an axis ratio 1:2.96 at a zero angle of attack were measured by Schubauer (1935, 1939). He found that there is a critical Re regime, extending from about Re= 85,000–312,000, based on the major axis of the cylinder. He also observed a socalled separation bubble upstream of the turbulent boundary layer at the critical Reynolds number. His main purpose, however, seems to have been investigation of the boundary layer characteristics. The drag coefficient was measured for an elliptic cylinder of axis ratio 1:2 at a = 0 and 90° by Delay and Sorensen (1953). All the characteristics of flow show that flow is subcritical at α = 90°, but at a = 0° they extend from the subcritical to the supercritical one. A series of experimental studies on steady and unsteady flow behaviors of elliptic cylinders were conducted by Modi and Wiland (1970), Modi and Dikshit (1971, 1975) and Modi and Ieong (1978). The mean and fluctuating pressure distributions, drag, lift, and moment coefficients, and also the near wake features were determined at the subcritical Reynolds number. They reported that the observed Re range corresponded to the subcritical flow regime and that the aerodynamic characteristics exhibited no dependency on Re. The lift and moment coefficients for an elliptic cylinder of axis ratio 2:5 indicated large variations with Re esp. at angles of attack smaller than 20° (Modi and Dikshit 1971). In an experimental investigation, Tiago et al. (2012) tested the effect of obstacle aspect ratio on tripped cylinder wakes. Heighttodiameter aspect ratios of 3 and 6 were considered. It was shown that a critical position exists for the tripwire, which is characterized in an abrupt change in the shedding frequency and wake structure. Results further suggested that the tripwire could strengthen 2D wake properties. The effect of the aspect ratio was described to be due to tipwake flow interactions and thus differs fundamentally from twodimensional geometries. For subcritical, turbulent wake flows of large aspect ratio cylinders, several studies have addressed the influence of the size and location of two symmetrically placed protrusions. These are typically tripwires mounted along the entire length of the cylinder at angular positions measured relative to the windward stagnation point. For Re in the order 10^{3}–10^{4}, wires of diameter 0.98–4.90 mm were found to have considerable effect on the wake when placed on the cylinder at azimuthal angles, α, ranging from 20° to 70° (Alam et al. 2003). It was found, for example, that for \(20^{^\circ } < \alpha < 40^{^\circ } ,\) the boundary layer reattaches downstream of the obstacle, but is thicker and can transition to turbulence, resulting in a changed shedding frequency and loading. For \(\alpha > 45^{^\circ }\), the boundary layer separates permanently for large wires, but for smaller wires can remain attached to \(\alpha \approx 70^{^\circ }\). Single tripwires have also been shown to significantly impact the shedding process. Zhou et al. (2007) investigated the effects of three different types of trip wires installed near the stagnation point on the flow properties around a circular cylinder. Each pair of wires was placed symmetrically at a distance from the cylinder. Their results showed that the drag coefficient decreased by 18% for Re = 200 and position \(\alpha = 40^{^\circ }\), and by 59% for \({\text{Re}} = 5.5 \times 10^{4}\) and position \(\alpha = 30^{^\circ }\). By studying the effect of trip wires on the vortexinduced vibrations, Hover et al. (2001) demonstrated that by using only a pair of wires at the 70° angle, the drag coefficient is reduced by up to about 50%, relative to the case in which no trip wires are used. Aydin and Ekmekci (2014) proposed estimation method at different Reynolds numbers and wire sizes. Igarashi (1986) experimentally investigated the effects of tripping wire on the transition in a boundary layer on a circular cylinder in a cross flow. Results show the roughness has no influence on the transition up to Re = 620 while at Re > 1220, it has full effect on the transition. Aiba et al. (1979) conducted an experimental study on the heat transfer and the flow around a circular cylinder with tripping wires. It is found that the heat transfer from the cylinder to the cross flow in very close connection with the width of near wake. Quadrante and Nishi (2014) examined the effect of tripping wires on the vibration of a circular cylinder subject to flows. They concluded that tripping wires positioned at 75° increased the amplitude of oscillation by 56%. While tripping wires positioned at 120° suppressed flow induced vibrations. Fukudome et al. (2005) estimated the performance of turbulence promoter for the symmetric airfoil at the high angle of attack by using the wind tunnel tests and numerical simulations. The results showed that the tripping wire near the leading edge improve the aerodynamic performances.
Studied Parameters
Turbulence intensity is an important parameter that characterizes the quality of fluid flow. It is expressed as:
where u′, u_{ref} and %Tu are the turbulent fluctuating velocity component (m/s), freeflow speed (m/s) and turbulence intensity percentage, respectively. Nondimensional velocity defect parameter is obtained as:
where U_{min} is the minimum velocity and w_{0} is the velocity defect. For an elliptic cylinder placed horizontally along the flow, blockage ratio is defined as the small diameter of the cylinder divided by the height of the gate of the wind tunnel. Previous studies have revealed that if a blockage ratio of < .08 is chosen for an elliptic cylinder, test error can be negligible. Considering the existing conditions in this research, the blockage ratio is β = .053.
Laboratory Equipment and Testing Method
One of the main tools for measuring instantaneous fluid flow velocity is the hotwire anemometer. Considering the unique features of this instrument, it is mainly used in the tests related to turbulent gas or airflows. The anemometer used in this research was of the constant temperature type, capable of measuring the average velocity, turbulence intensity and frequency of vortices shed from behind the elliptic cylinder. The wind tunnel employed in these tests was an opencircuit blowdown wind tunnel that used a 7 kW power motor to create airflow with a velocity of 30 m/s. The test section of this wind tunnel was made of Plexiglas and its length, width and height were 168, 40 and 40 cm, respectively. Figure 1 shows the free stream turbulence intensity values in the test chamber, in the absence of any elliptic cylinder, for two flow velocities. The onedimensional probe used in the tests had a sensor of 1mm length and 5µm diameter. A precise mechanism with 3 degrees of freedom was used to move the probe to different points. The precision of this probe displacement mechanism was .01 mm. A schematic of the wind tunnel is shown in Fig. 2.
A Schematic Layout of the Wind Tunnel
In this study, the wake produced behind a smooth elliptic cylinder at an angle of attack of 0° (aligned with the X axis) and in the presence of trip wires was analyzed. The cylinder used in the tests was made of aluminum, and had a length of 390 mm, major diameter of 42.4 mm and minor diameter of 21.2 mm. A schematic of the elliptic cylinder is presented in Fig. 3 (A = 42.4 mm, B = 21.2 mm, AR = A/B = 2).
The data collecting and measuring stations were positioned according to the ratio of the distance from the elliptic cylinder to the elliptic cylinder’s small diameter, as: X/B = 2.5, 5, 10, 15, 20 and 25. Each pair of turbulence promoter wires with .5 mm diameter was installed symmetrically at two different positions at angles of 23.7° and 40.9° relative to the stagnation point. The elliptic cylinder was tested at Re = 38,550 and 64,250 for air velocities of 15 m/s and 25 m/s, respectively.
Numerical Analysis, Governing Equations and Boundary Conditions
Realizable k–ε turbulence model (2D, steady, incompressible) was used for the numerical analysis and the SIMPLEC algorithm was employed for pressure–velocity coupling. This algorithm is used for complex flows with turbulence or additional physical models and improves convergence. Standard scheme was used for pressure discretization, whereas for momentum and dissipation of turbulence kinetic energy, the secondorder upwind scheme was utilized and convergence criterion was set at 10^{−6}. Equations for continuity, momentum in the Xdirection, turbulence kinetic energy and dissipation are as expressed in Eqs. (3)–(6). Velocity inlet condition and outflow were considered at the inlet and outlet, respectively, and the elliptic cylinder, trip wire and other surfaces were assumed as no slip wall.
Figure 4 shows a view of generated grid for study of behavior and characteristics of the wake produced behind an elliptic cylinder at 0° angle of attack, in the presence of trip wire. Turbulence trip wires of .5 mm diameter were installed symmetrically on the elliptic cylinder at angle of 23.7° relative to the stagnation point. Figure 5 shows the y^{+} (y plus) values on walls of the elliptic cylinder. The values of y plus (y^{+}) are lower than 30.
Grid Independency
Effect of number of grid points on the drag coefficient is shown in Table 1. The results indicate that a 150 × 75 grid is adequate to achieve grid independency for the current problem.
Validation
Lindsey (1938) performed comprehensive studies on drag coefficients of various objects. Among these tests, the results for a smooth elliptic cylinder with a diameter ratio of ½ with an angle of attack equal to 0° for various Re numbers are presented in Fig. 6.
Since, to the best of our knowledge, there has been no study on the impact of trip wires on the wake of an elliptic cylinder, validation was only possible by Fig. 6, which presents data for smooth elliptic cylinders.
Analysis and Discussion
Distribution of Nondimensional Average Velocity Profiles
Distribution of the profile of the dimensionless mean velocity for the smooth elliptic cylinder is shown in Fig. 7. Considering the transient and varying nature of oscillating velocity components, velocity profiles are symmetrical during one period. Vortices form instantaneously and asymmetrically behind the elliptic cylinder; however, in view of Fig. 7, even at the positions near the elliptic cylinder, the plotted average velocity profiles are completely symmetrical.
The reason can be attributed to the fact that the formation of vortices behind the cylinder is instantaneous and that the vortices recur reciprocally during their period. Now, if the data sampling time at a point is greater than the formation period of vortices, by calculating the timeaveraged velocity at the considered points, the average velocity at each point is obtained, which does not represent the asymmetric effects of instantaneous velocities. In Fig. 7, the data sampling time is almost 500 times the formation period of vortices.
Another point of interest in Fig. 7, is the existence of two extremum regions in the velocity profiles of cross sections near the elliptic cylinder. This can be attributed to the momentum of the boundary layer formed on the cylinder surface. At cross sections near the elliptic cylinder, the existing momentum in the boundary layer formed on the surface increases the energy of the adjacent points after the disintegration of the boundary layer and ultimately leads to their increased velocity. This effect is no longer present at cross sections farther behind the elliptic cylinder, and the momentum is no longer able to alter the velocity of fluid particles.
The other noteworthy point is that at distances farther from the back of the cylinder, the velocity difference of the inside and outside of the wake decreases and the wake width gets larger, which consequently results in a more uniform average velocity profile. On the other hand, at near distances, due to separation and return flows, flow dissipation is higher and as we move away from the cylinder, this dissipation is decreased and the flow inside the wake becomes more uniform, as is evident in Fig. 5, where the curves corresponding to the two Re numbers approach each other very closely from station X/B = 5 onward. This is due to the fact that vortex dissipation rate is higher at Re = 38,550 than Re = 64,250. Also, from this position onward, the flow behavior changes, which includes the elimination of return flows, flows becoming more uniform, and the weakening of flow separation effects. This leads to gradual velocity increases within the wake from the initial to the end data collecting stations, thereby altering the average velocity profiles and converting them to relatively uniform lines at the end stations.
The flow passing over the surface of a smooth elliptic cylinder (with no trip wires) forms a boundary layer that later separates from the surface. This separation leads to the formation of a region called wake behind the elliptic cylinder that has different characteristics from those of a free stream. Conversely, when flow encounters a bump along the way, significant changes occur in the characteristics of the boundary layer as well as the wake. A trip wire installed on the surface of the cylinder acts as a bump for the passing flow and causes a phenomenon known as boundary layer transition. In other words, streamlines detach from the cylinder surface by striking the trip wire and deflecting upward, and come back again to the surface at a downstream point called the reattachment point. This point is the starting point of boundary layer reestablishment. When boundary layer forms again, there is a separation point as well. Due to the presence of trip wires, the boundary layer forms on the cylinder surface at a further downstream point compared to the case of smooth (unwired) cylinder. Therefore, the wake that forms has a smaller width.
The trip wires initiate major changes in the mean velocity profiles. It should be noted, however, to avoid an excessive number of graphs, only the flow diagrams associated with Re = 64,250, which has a higher effect on flow characteristics than Re = 38,550, have been presented for trip wires at each position. The positive effect of trip wires on flow characteristics in the wake behind the elliptic cylinder increases the mean velocity. The installation location and position of trip wires could have an either positive or negative effect on flow characteristics. When wire position causes an increase in the width of the wake, powerful eddies are created behind the elliptic cylinder. In this case, the average velocity is reduced in the wake, and flow turbulences are intensified. Thus, the conditions for improving the flow characteristics deteriorate. However, if, under the effect of trip wires, the wake width gets smaller, the vortices behind the elliptic cylinder weaken, and the flow characteristics improve (Alam et al. 2003). Figure 8 shows the distribution of the profile of the dimensionless mean velocity at different positions.
Among the nondimensional average velocity profiles shown in Fig. 8, for the station X/B = 2.5, it is observed that at sections near the elliptic cylinder, the velocity peak in the profile related to the smooth cylinder is higher than that in other the profiles. The reason is that the boundary layer created on the smooth cylinder has some momentum that, after dissipation, helps increase the energy, and thus the velocity of the adjacent points. This effect is no longer present at the stations farther away behind the elliptic cylinder and therefore, there is no peak in the velocity profiles of the next stations; whereas, for the trip wires at position \(\theta = 40.9^{^\circ }\), this velocity peak has much less intensity. Hence, upon striking the trip wire, the flow loses some of its velocity and thereby its momentum. It can therefore be deduced that the establishment of a reattachment point for the boundary layer wastes some of the flow momentum in the boundary layer. Another point is that when the separation of boundary layer on surface occurs at a point farther away from the upstream, a wider wake is formed, creating more powerful eddies, in which flow velocity is lower than that in the wakes with smaller widths. This can be seen more clearly in Fig. 8. As indicated, velocity has a higher value in the profile associated with wire position \(\theta = 40.9^{^\circ }\) than in other profiles, and thus has a greater impact on flow characteristics. What can also be concluded is that as the distance from the elliptic cylinder increases, the wake becomes wider and the difference between flow velocities in and out of the wake becomes smaller. This is illustrated in Fig. 8. The reason that the wake behind the elliptic cylinder diminishes at farther distances from the back of model to the point where there is no wake is that the shear layer of the existing wake is affected by the free stream, which tries to diminish the shear layer of the wake.
Nondimensional Velocity Defect Profile
The difference between the free stream velocity and the minimum velocity is that, the wake has a decreasing rate for successive stations, starting from the one near the elliptic cylinder to the station in a far downstream region. This velocity difference is called velocity defect. As observed in Fig. 9, the gradients of the velocity defect profiles are the same for both diagrams, and from station X/B = 5 onward, these two graphs overlap. This shows that from this point on, the velocity profiles approach each other very closely; which is demonstrated in Fig. 7. Additionally, it is shown that at the two preceding stations in Fig. 9, the gradient for the graph corresponding to Re = 38,550 is greater than that for the graph associated with Re = 64,250.
The severe gradient in the velocity defect profile indicates the velocity variation rate (\(\frac{\partial }{\partial x}\left( {\Delta u} \right)\)). This parameter also depends on the energy dissipation rate. Therefore, the dissipation rate in this profile should be higher than the one in the other profile. This can be realized from Fig. 7, in which the minimum value in the velocity profile for Re = 38,550, measured at stations far from the elliptic cylinder, coincides with that of the other velocity profile for Re = 64,250. Due to the higher average velocity in the wake for the .5 mm diameter trip wire compared to the other velocity profiles, it is necessary to have a severe drop in the velocity defect graph in Fig. 10 for the .5 mm wire (θ = 23.7° and θ = 40.9°) in order for this graph to be lower than the other graphs. The reason is that, due to the large gradient of the velocity profile in this case, dissipation rate is higher for the stations near the elliptic cylinder. Thus, by getting away from the elliptic cylinder, a severe drop occurs in the dimensionless velocity defect graph for this wire. Another point is that the closer the velocity defect curves get to the 1/X curve, the more optimal the conditions will be. This is due to the fact that for a graph in the form of 1/X, the velocity difference in the wake is minimum, which in turn maximizes the flow velocity in the wake.
Wake width for the flow around the cylinder at Re = 38,550 and 64,250 is shown in Fig. 11. As can be seen, wake width has an ascending trend. From X/B = 2.5 to X/B = 10, wake width at Re = 38,550 is larger than that of Re = 64,250. However, at the vortex shedding region (end stations), this value is approximately similar for the two cases. It should be noted that increased wake width leads to a larger shear stress region. Figures 12 and 13 depict the variations of wake width in terms of different angles for the two Re values. It is evident from Fig. 12 that the wake width profile is higher than the other profiles at \(\theta = 40.9^{^\circ }\), resulting in a larger wake. In other words, this value has greater growth compared to the other states. Therefore, it cannot be expected that the end stations have smaller wake widths compared to the other stations if the wake width near the elliptic cylinder has the minimum value. As a result, the formation of wake boundary for each state is strongly dependent on the installation location of the wire and Re.
Dimensionless Turbulence Intensity Profiles
Distribution of the dimensionless turbulent intensity for the smooth cylinder is shown in Fig. 14. The direct relationships of energy dissipation with average velocity gradient \({\raise0.7ex\hbox{${\partial \bar{u}}$} \!\mathord{\left/ {\vphantom {{\partial \bar{u}} {\partial y}}}\right.\kern0pt} \!\lower0.7ex\hbox{${\partial y}$}}\) and the gradient of velocity’s fluctuating component \({\raise0.7ex\hbox{${\partial u^{\prime}}$} \!\mathord{\left/ {\vphantom {{\partial u^{\prime}} {\partial y}}}\right.\kern0pt} \!\lower0.7ex\hbox{${\partial y}$}}\), respectively, shown in Figs. 7 and 14, indicate that the wake flow at Re = 38,550 decays more quickly. Hence, from station X/B = 5 onward, due to the dissipation of energy, the average and turbulent velocity profiles associated with Re = 38,550 get closer to and coincide with the velocity profiles corresponding to Re = 64,250. It is also observed that at larger distances from the elliptic cylinder, fluctuations of the dimensionless turbulence intensity gradient weaken drastically.
In Fig. 14, two extremum points are observed in the dimensionless turbulence intensity profile at Re = 64,250 for station X/B = 2.5; which, due to the flow becoming more uniform, are eliminated the further they get from the back of the elliptic cylinder. This phenomenon emerges as a result of the vortex flows and eddies near the cylinder, and uniform flows away from the cylinder.
In view of Figs. 7 and 14, a similar behavior is observed at each station. When both of the average velocity profiles almost overlap with each other at station X/B = 5, the two turbulence intensity profiles are also roughly coincident, exactly at the same station. This is while the magnitude of average velocity in the wake is increasing in Fig. 7, but the turbulences in the wake are declining in Fig. 14. The reason for this is the impact of energy dissipation on flow characteristics; so that the increase of average flow velocity \(\bar{u}\) and reduction in flow turbulences \(u^{\prime}\) in the wake occur at a uniform rate, with the same trend.
Another point of interest is that the peak of flow turbulences in the turbulence intensity profile in the wake region behind the elliptic cylinder indicates the conditions of a wake point where most flow turbulences exist. It is also observed that by plotting the velocity profile and multiplying flow turbulences in a diagram, the peak of turbulences falls on the turning point of the velocity profile. Distribution of the dimensionless turbulent intensity at different positions is shown in Fig. 15. It indicates that the width of the turbulence profile for the smooth cylinder is greater than that for the other profiles. This is due to the fact that the separation point for the smooth elliptic cylinder occurs further away from the flow upstream; thus, larger eddies are generated and the width of its turbulence profile is greater. This also occurs for the velocity profile in Fig. 8. As seen in Fig. 13, the turbulence intensity profile for position \(\theta = 40.9^{^\circ }\) is smaller than the other turbulence profiles, and the wake associated with the tripwired cylinder disintegrates faster at this position than at the other positions, causing the turbulence peaks to vanish more quickly. In this case, the flow wake has a smaller width and its vortices are weaker, and therefore it vanishes faster than the other cases. To make the wake width smaller under the effect of trip wire, a reattachment point should occur further away downstream to form a boundary layer. In view of Fig. 15, this point is better established at position \(\theta = 40.9^{^\circ }\) than at other positions; because the velocity difference (i.e., velocity defect) in the wake is less at this position than at the others.
Numerical Analysis of Impact of Various Reynolds Numbers on Velocity Profile for Elliptic Cylinder with Trip Wires
Variations in mean velocity profile at different Re numbers in terms of X/B = 2.5, 5, 10, 15, 20 and 25 can be seen in Fig. 16. The results indicate that an increase in Re (higher input airflow velocity) results in critical and important changes to the mean velocity profile. As shown in Fig. 16, velocity profiles are symmetric in a period due to the variable and nonpermanent nature of the fluctuating components of velocity. Although vortices form instantaneously and unsymmetrically behind the model, the mean velocity profiles are completely symmetric even at the first stations near the cylinder. The reason could be found in the fact that formation of vortices behind the model is instantaneous and is repeated during their period. Now, if the recording time at a point is higher than the vortex formation time, mean velocity at the points in question can be found by calculating the time average value of velocity, which does not exhibit the effects of the unsymmetrical nature of instantaneous velocity. As seen, at longer distances from the back of the elliptic cylinder, velocity difference between the inside and outside of the wake decreases and wake width increases, leading to a more uniform mean velocity profile. At the stations near the elliptic cylinder, due to separation and return flows, flow dissipations increase; whereas at longer distances, dissipations reduce and the inside of the wake becomes more uniform. On the other hand, Fig. 16 shows that mean velocity profiles for all Re numbers from X/B = 5 onward are very close to one another. This is because vortex dissipation rate due to the contact between the vortex and the shear layer of the wake is higher at Re = 38,550 than other Re values.
At the first station, due to the low pressure, there is a large pressure difference with reference to potential flow. In this state, variations in velocity from the wake to the potential flow at the top and bottom of the elliptic cylinder are high. In other words, velocity has a large gradient in these two regions. At cross sections near the model, the existing momentum in the boundary layer formed on the cylinder surface causes a rise in the energy of the adjacent points, and in turn their velocity, as the boundary layer vanishes. This phenomenon loses its effect at the stations far downstream and becomes unable to alter the velocity of fluid particles.
In regions near the wake (X/B ≪ 2.5), profiles in the rotating region right after the model are V shaped whereas, they are U shaped downstream of the rotating region. U shaped profiles in the vortex shedding region (X/B ≫ 10) can lead to intensified turbulence and increased mixing. The minimum velocity values inside the wake at stations X/B = 2.5 and 5 are lower than at Re numbers other than Re = 38,550 and reach an approximately similar value at the end stations. There, variations in mean velocity profile for both Re numbers are similar which can be attributed to the changes in flow behavior including vanishing of return flows, flow becoming uniform and reduction in flow separation effects.
Modifying the Schlichting’s Velocity Defect Profile
Schlichting (1968) expressed a relationship for the wake of twodimensional objects at far away stations where static pressure of wake is equal to their value outside the wake. Based on this relationship, some equivalents for velocity profile can be obtained in terms of velocity defect with respect to the half width. The main problem of this equation is its failure to properly describe wake for stations close to the elliptic cylinder. Accordingly, by adding two coefficients, it can be optimized such that it can, with excellent accuracy, describe wake for all stations. The final equation has the following form:
where d is the large diameter of the elliptic cylinder. In this equation, the S_{1} correction coefficient changes the velocity profile in the Y direction and the S_{2} correction coefficient changes the velocity profile in the X direction. Meantime, the \(S_{n}\) factor is defined as:
where a can be changed based on the type of wire, its condition and current situations. In Fig. 17, a comparison is performed between the experimental results of velocity profile of the circular cylinder with a diameter of 2 cm with Angrili et al. (1982) at the station X/D = 2.5 and Re = 24,200, and an optimized equation for it is presented as:
where U_{ref} and u are freeflow velocity (m/s) and wake flow velocity (m/s), respectively. In the above equation, the a values obtained for S_{1} and S_{2} are 750 and 37, respectively.
Analysis of the Drag Coefficient
Equations used for measurement of drag force are derived by applying momentum and conservation of mass equations in a control volume. There are two types of drag. One is the pressure drag, which comprises the majority of the total drag force in 2D objects. Employing trip wires improves flow conditions such that the pressure difference between the stagnation points at the front and back of the elliptic cylinder is reduced. Therefore, by using trip wires and consequently formation of the reattachment point, wake width decreases, which facilitates a pressure rise behind the elliptic cylinder. This increased pressure results in the reduction in the pressure difference around the elliptic cylinder, which in turn, reduces the force exerted on it caused by the difference in pressure. Furthermore, Khan et al. (2005) showed that for a smooth elliptic cylinder, the pressure drag coefficient is expressed as:
where \(C_{{D_{p} }}\) is the pressure drag coefficient, C_{p} is the pressure coefficient, \(e = \sqrt {1  AR^{2} }\) is the ellipse’s eccentricity and \(AR = {\raise0.7ex\hbox{$b$} \!\mathord{\left/ {\vphantom {b a}}\right.\kern0pt} \!\lower0.7ex\hbox{$a$}}\) is the ratio of the axes where b and a are half of the minor and major diameters of the ellipse, respectively. Frictional drag is the other form of drag, which is created due to the viscous shear forces on the cylinder surface after boundary layer formation, as expressed in Eq. (11). In this equation, \(C_{{D_{\text{f}} }}\) is the friction drag coefficient and \(C_{\text{f}}\) is friction coefficient.
Lu and Bragg (2002) conducted numerous studies on parameters affecting the drag coefficient. They also analyzed the impacts of flow turbulence and oscillations and obtained valuable results. Van Dam (1999) derived an equation for calculation of drag coefficient in which, there were Reynolds stress and turbulence intensity terms. Variations of fluid density and viscous term \(\mu {\raise0.7ex\hbox{${\partial u}$} \!\mathord{\left/ {\vphantom {{\partial u} {\partial x}}}\right.\kern0pt} \!\lower0.7ex\hbox{${\partial x}$}}\) were however neglected. The components of equation are expressed as Eq. (12). In this equation, U_{ref} is freeflow speed (m/s) and \(\bar{u}\) is mean velocity (m/s).
The first component corresponds to the pressure term:
P_{s,e} is static pressure outside the wake (model upstream), \(P_{\text{s,w}}\) is static pressure inside the wake. The second component corresponds to the momentum term:
L is characteristic length of the model (cross section height of the cylinder). The third component corresponds to the Reynolds stress term:
On the other hand, because the exact calculation of static pressure within the wake is a difficult task, based on several assumptions, the most important of which is that the mean velocity \(\bar{v}\) and \(\bar{w}\) are very small and negligible and that the total pressure is constant along the streamline, the equation becomes fairly simple. As a result, the original Van Dam equation takes the ultimate form of:
where \(\overline{\acute{q}} = \rho \left( {\overline{{\acute{u}^{2} }}} + \overline{{\acute{v}^{2} }} + \overline{{\acute{w}^{2} }} \right)/2\) and it is assumed that far away downstream, the flow is homogenized and therefore u′ = v′ = w′. On the other hand, \(\bar{q} = \frac{1}{2}\rho \bar{V}^{2}\) is the timeaveraged dynamic pressure and \(q_{\text{ref}} = \frac{1}{2}\rho V^{2}\) is the dynamic pressure. The method presented by Van Dam, based on its assumptions, is limited to steady flow and contains errors for unsteady flow. On the other hand, the frictional drag coefficient in terms of axis ratio AR = b/a is presented as Khan et al. (2005):
where \({\text{Re}}_{{\mathcal{L}}} = {\raise0.7ex\hbox{${{\mathcal{L}}U_{\text{ref}} }$} \!\mathord{\left/ {\vphantom {{{\mathcal{L}}U_{\text{ref}} } \nu }}\right.\kern0pt} \!\lower0.7ex\hbox{$\nu $}}\) is the Reynolds number in which \({\mathcal{L}}\) is equal to the major diameter of ellipse \(\left( {2a} \right)\). The pressure drag coefficient is presented as:
Eventually, the total drag coefficient is equal to the sum of pressure and friction drag coefficients as:
Comparison of the results obtained by Eq. (15) and the current work yields errors equal to .65% and 5.2% for Re = 38,550 and Re = 64,250, respectively. Variations of the drag coefficient for the smooth cylinder at different stations are shown in Fig. 18. As can be seen, the drag coefficient for Re = 38,550 is slightly larger than the other Re. This is because as separation is delayed, wake width decreases and since the space that the wake occupies reduces as well, pressure increases behind the elliptic cylinder, which results in reduced pressure difference around it (Behara and Mittal 2011). Whereas, \(C_{p} = \frac{{p_{1}  p_{2} }}{{\frac{1}{2}\rho U_{\text{ref}}^{2} }}\) in Eq. (9) decreases as this pressure difference reduces. Because, according to Khan’s study (Khan et al. 2005), P_{1} – P_{2} is the pressure difference between the upstream and downstream of the elliptic cylinder. Therefore, Eq. (6) also indicates that reduction in pressure difference leads to a decrease in the drag coefficient. It should however be noted that for a smooth elliptic cylinder, drag coefficient does not vary significantly. As it is clear in Fig. 19, the drag coefficient for the .5 mm diameter wire (θ = 23.7° and 40.9°) is less than all the other cases. Considering the mean velocity charts in Fig. 8, it is clear that the velocity in the wake for the .5 mm wire is more than the others. On the other hand, in Fig. 15, it is clear that turbulence for this wire is less than others. Thus it can be deduced that reducing the drag coefficient results in increasing the amount of mean velocity and reducing the flow turbulences.
Table 2 shows the variations of drag coefficient at all the positions in terms of X/B for the two Re values. The results indicate that the most optimum value of the drag coefficient for Re = 38,550 occurs at \(\theta = 40.9^\circ\) from the wire installation location which is 55% less than the value corresponding to the smooth cylinder. In addition, for Re = 64,250, the optimal drag coefficient occurs at \(\theta = 40.9^\circ\) which shows a 75% reduction compared to the smooth cylinder case. For the installation angle of \(\theta = 23.7^\circ\), drag coefficient decreases by 40% at Re = 64,250. It can be inferred from these results that the effect of wire on reducing the drag coefficient is more evident for the case where Re = 64,250 than the case where Re = 38,550.
Kolmogorov Length Scale
Microscales are the smallest scales in turbulent flow. At the Kolmogorov scale, viscosity dominates and the turbulent kinetic energy is dissipated into heat. They are defined by Kolmogorov microscales, as defined by Eq. (16–18).Where ɛ is the average rate of dissipation of turbulence kinetic energy per unit mass and ϑ is the kinematic viscosity of the fluid.
Typical values of the Kolmogorov length scale, for atmospheric motion in which the large eddies have length scales in the order of kilometers, range from .1 to 10 mm. For smaller flows such as the ones in laboratory systems, \(\eta\) may be much smaller. The Kolmogorov length scales at the different station are shown in Fig. 20. Figure 21 also shows contours of x velocity for different Re numbers at different positions. The reason for this phenomenon can be found in the fact that the separation point for the smooth elliptic cylinder occurs upstream of the flow (front of the elliptic cylinder) and thus leads to larger wakes. Whereas, for the position \(\theta = 40.9^{^\circ }\), the wake corresponding to the wired cylinder disintegrates earlier than the other cases and its vortices are weaker than them, leading to their earlier vanishing. Upon entering the wake, streamlines fall to a lowpressure region, which results in formation of lowpressure vortices. Growth of these vortices and continuation of their spinning are due to the shear layer of the wake and influx of streamlines to the wake. They are broken into smaller vortices as they get away from the model due to energy dissipation and finally, Kolmogorov eddies are formed which directly transform to heat as energy dissipates.
Conclusion
In the present work, the effect of trip wire on the wake characteristics of a smooth elliptic cylinder at a zero angle of attack was investigated. An opencircuit wind tunnel and blower were employed to create the required airflow, and the data collection stations were established at 6 different locations.

Results indicated that a turbulence promoter wire has a significant effect on flow characteristics and reduction in the drag coefficient, and that this influence strongly depends on the location where the trip wire is installed on the model.

The findings show that for a smooth cylinder, drag coefficient diminishes by increasing Re.

For the smooth cylinder, the presence of a severe gradient in the dimensionless average velocity profile and turbulence intensity profile at Re = 38,550, is indicative of the fact that the energy dissipation rate is higher at Re = 38,550. This speeds up the disintegration of vortices created behind the elliptic cylinder in this flow condition.

A severe gradient in the average velocity profile is directly related to the strength of vortices generated behind the elliptic cylinder.

The reduction in drag coefficient is accompanied by the increase of average velocity in the wake and also, the reduction in flow turbulences.

The structure of turbulence intensity profile is such that the turbulence peaks at stations near the model pass through the turning point of the velocity profile. In other words, where the flow has its highest turbulence value, the direction of the velocity variation rate also changes.
Uncertainty of the Results
The maximum error in the measurement of the instantaneous velocity values is discussed hereafter. The parameters affecting instantaneous velocity and how much they affect it are presented below.
The effect of each parameter of the velocity is obtained from the following equation, which is called the relative standard uncertainty:
where k is the convergence coefficient and \(\frac{1}{U} \cdot \Delta y_{i}\) is the standard deviation.
The total error stemming from the effect of various parameters on instantaneous velocity is obtained as:
Calibration Error
The calibration error depends on whether calibration was conducted by a special device or a Pitot tube can be a major source of uncertainty. Since a Pitot tube and a manometer were used in the current study, the error was 1% at max.
Curve Fitting Error
The maximum curve fitting error was 1%.
Analog to Digital Conversion Error
The converter used in the study was a 12bit board with a maximum voltage of 10 V and maximum error of .12%.
Probe Placement Error
If the angle change of the probe does not exceed 1°, this error can be neglected.
Temperature Variations Error
This error is comprised of two parts:

(A)
The temperature variations of maximum 1 °C during calibration led to an error of .4%.
$$\begin{aligned} {\text{Error}}\left( \% \right) & = \frac{ 1}{k} \cdot \frac{ 1}{U} \cdot \Delta y_{i} = \frac{ 1}{k} \cdot \frac{ 1}{U} \cdot \frac{\Delta T}{{T_{\text{w}}  T_{\text{o}} }} \cdot \left( {\frac{A}{B} \cdot \frac{ 1}{\sqrt U } + 1} \right) = . 0 0 0 2 6 9\\ \Delta T & = 1^{^\circ } ,\,\,T_{\text{w}}  T_{\text{o}} = 2 0 0\,^{ \circ } {\text{C}},\,\,A = 1 / 3 9 6,\,\,B = 0 / 8 9 5,\,\,k = \sqrt 3,\,\,U = 1 5\,\,{\text{m/s}} \\ \end{aligned}$$(23) 
(B)
Temperature variations during the test: The variations during the test were up to 2 °C resulting in an error of .4%.
$$\begin{aligned} {\text{Error}}\left( \% \right) & = \frac{ 1}{k} \cdot \frac{\Delta T}{ 2 7 3} = 0/ 4 2\\ \Delta T & = 2^{^\circ } ,{\mkern 1mu} \,\,k = \sqrt 3\\ \end{aligned}$$(24)
Ambient Pressure and Humidity Error
Since at the site of the wind tunnel facility, ambient pressure and humidity were constant, the error associated with these values can be neglected and therefore considered equal to zero.
Considering all the abovementioned errors, the total error of the instantaneous velocity measurement was obtained from Eq. (25) and is equal to 4.03%.
Abbreviations
 AR:

Axes ratio (proportion of small diameter to large diameter of elliptic)
 B :

Small diameter elliptic (m)
 b _{1/2} :

Half width (m)
 \(C_{{D_{\text{f}} }}\) :

Friction drag coefficient
 \(C_{{D_{p} }}\) :

Pressure drag coefficient
 \(C_{\text{D}}\) :

Drag coefficient
 \(C_{\text{f}}\) :

Friction coefficient
 C _{ p } :

Pressure coefficient
 e :

Eccentric elliptic
 L :

Characteristic length of the model (m)
 P _{1} :

Front stagnation point pressure (kg/m/s^{2})
 P _{2} :

Rear stagnation point pressure (kg/m/s^{2})
 P _{s,e} :

Static pressure outside the wake (model upstream) (kg/m/s^{2})
 P _{s,w} :

Static pressure inside the wake (kg/m/s^{2})
 Re:

Reynolds number
 %Tu:

The percentage of turbulence intensity
 U _{ref} :

Freeflow speed (m/s)
 u :

Wake flow speed (m/s)
 u _{ i } :

Horizontal component of freeflow speed (m/s)
 \(\bar{u}\) :

Mean velocity (m/s)
 \(u^{\prime}\) :

Turbulent velocity fluctuation component (m/s)
 w _{0} :

Velocity defect parameter (m/s)
 X :

Distance from the rear stagnation point (m)
 β :

Blockage ratio
 \(\theta\) :

Installation angle of wire
 μ :

The dynamic viscosity (kg/m/s)
 ρ :

Density (kg/m^{3})
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Yadegari, M., Bak Khoshnevis, A. Numerical and Experimental Study of Characteristics of the Wake Produced Behind an Elliptic Cylinder with Trip Wires. Iran J Sci Technol Trans Mech Eng 45, 265–285 (2021). https://doi.org/10.1007/s40997020003736
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Keywords
 Trip wire
 Drag coefficient
 Elliptic cylinder wake
 Wind tunnel
 Numerical analysis