Global linear stability analysis of time‐averaged flows

The global linear stability analysis (LSA) of stationary/steady flows has been applied to various flows in the past and is fairly well understood. The LSA of time‐averaged flows is explored in this paper. It is shown that the LSA of time‐averaged flows can result in useful information regarding its stability. The method is applied to study flow past a cylinder at Reynolds number (Re) beyond the onset of vortex shedding. Compared with the direct numerical simulation, LSA of the Re=100 steady flow severely underpredicts the vortex shedding frequency. However, the LSA of the time‐averaged flow results in the correct value of the non‐dimensional frequency, St, of the associated instability. Copyright © 2007 John Wiley & Sons, Ltd.     

Free v/s forced vibrations of a cylinder at low Re

Free vibrations of a circular cylinder of low non-dimensional mass are investigated at low Reynolds numbers. Computations are carried out for 5% blockage. Lock-in is observed for a range of Re and is accompanied with hysteresis at both lower as well as higher Re ends of the synchronisation/lock-in region. It is well known that the lock-in regime for free vibrations depends on the non-dimensional mass of the oscillator. The results from the present computations are compared with the data for forced vibrations from Koopmann (Journal of Fluid Mechanics, 28, 501–512, 1967) on a Y _max/D vs. f* plot, where Y _max is the maximum oscillation amplitude and f* is the ratio of cylinder vibration frequency to the vortex shedding frequency for a stationary cylinder. Good agreement is observed for the critical amplitude needed for onset of synchronisation between the forced and free vibrations. The results from the free vibrations are compared to the predictions from the linear oscillator model by assuming that the forces on the cylinder are unaffected as a result of vibrations. It is found that, for low mass oscillators, the modification of vortex shedding frequency and lift coefficient due to cylinder oscillations leads to the enhancement of the lock-in regime.     

Effect of blockage on free vibration of a circular cylinder at low Re

The effect of the blockage on vortex‐induced vibrations of a circular cylinder of low non‐dimensional mass (m^*=10) in the laminar flow regime is investigated in detail. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables form in two dimensions. The transverse response of the cylinder is found to be hysteretic at both ends of synchronization/lock‐in region for 5% blockage. However, for the 1% blockage hysteresis occurs only at the higher Re end of synchronization/lock‐in region. Computations are carried out at other blockages to understand its effect on the hysteretic behavior. The hysteresis loop at the lower Re end of the synchronization decreases with decrease in blockage and is completely eliminated for blockage of 2.5% and less. On the other hand, hysteresis persists for all values of blockage at the higher Re end of synchronization/lock‐in. Although the peak transverse oscillation amplitude is found to be same for all blockage (～0.6D), the peak value of the aerodynamic coefficients vary significantly with blockage. The r.m.s. values show lesser variation with blockage. The effect of streamwise extent of computational domain on hysteretic behavior is also studied. The phase between the lift force and transverse displacement shows a jump of almost 180° at, approximately, the middle of the synchronization region. This jump is not hysteretic and is independent of blockage. Copyright © 2008 John Wiley & Sons, Ltd.     

Spanwise correlations in the wake of a circular cylinder and a trapezoid placed inside a circular pipe

The spanwise correlation of a circular cylinder and a trapezoidal bluff body placed inside a circular pipe in fully developed turbulent regime is studied using hotwire anemometer. The present configuration possesses complex fluid structure interaction owing to the following features: high blockage effect; low aspect ratio of the body; upstream turbulence and interaction of axisymmetric flow with a two dimensional bluff body. The spatial correlation of such configuration is seldom reported in the literature. Results are presented for Reynolds number of Re_D=1×10⁵. Three different blockage ratios (0.14, 0.19 and 0.28) are considered in the present study. Correlation coefficient is observed to improve with increase in blockage ratio. Compared to a circular cylinder, a trapezoidal bluff body possesses high correlation length. The near wall effects tend to increase the phase drift, which is reflected in low correlation coefficients close to the pipe wall. The results show that the simultaneous effect of curvature, low aspect ratio and upstream turbulence reduces the correlation coefficients significantly as compared to unconfined and confined (parallel channel) flows. The low frequency modulations with a circular cylinder are higher for lower blockage ratios. The three-dimensionality of vortex shedding for trapezoid with a blockage ratio of 0.28 was observed to be lower compared to circular cylinder and all other blockage ratios. Low frequency modulations were found to be responsible for weak vortex shedding from a circular cylinder compared to a trapezoidal bluff body. The vortex shedding is observed to be nearly two dimensional in case of a trapezoidal bluff body of blockage ratio 0.28.     

A finite element formulation for global linear stability analysis of a nominally two‐dimensional base flow

A stabilized finite element method, to carry out the linear stability analysis of a two‐dimensional base flow to three‐dimensional perturbations that are periodic along span, is presented. The resulting equations for the time evolution of the disturbance requires a solution to the generalized eigenvalue problem. The analysis is global in nature and is also applicable to non‐parallel flows. Equal‐order‐interpolation functions for velocity and pressure are utilized. Stabilization terms are added to the Galerkin formulation to admit the use of equal‐order‐interpolation functions and to eliminate node‐to‐node oscillations that might arise in advection‐dominated flows. The proposed formulation is tested on two flow problems. First, the mode transitions in the circular Couette flow are investigated. Two scenarios are considered. In the first one, the outer cylinder is at rest, while the inner one spins. Two linearly unstable modes are identified. The primary mode is real and represents the axisymmetric Taylor vortices. The second mode is complex and consists of spiral vortices. For the counter‐rotating cylinders, the primary transition is via the appearance of spiral vortices. Excellent agreement with results from earlier studies is observed. The formulation is also utilized to investigate the parallel and oblique modes of vortex shedding past a cylinder for the Re = 100 flow. It is found that the flow is associated with a large number of unstable oblique shedding modes. The parallel mode of vortex shedding is a special case of this family of modes and is associated with the largest growth rate. Copyright © 2014 John Wiley & Sons, Ltd.     

Issue Information

Three‐dimensional direct numerical simulation results of flow past a circular cylinder are influenced by numerical aspects, for example the spanwise domain length and the lateral boundary condition adopted for the simulation. It is found that inappropriate numerical set‐up, which restricts the development of intrinsic wake structure, leads to an over‐prediction of the onset point of the secondary wake instability (Re_cr). A best practice of the numerical set‐up is presented for the accurate prediction of Re_cr by direct numerical simulation while minimizing the computational cost. The cylinder span length should be chosen on the basis of the intrinsic wavelength of the wake structure to be simulated, whereas a long span length is not necessary. For the wake transitions above Re_cr, because the wake structures no longer follow particular wavelengths but become disordered and chaotic, a span length of more than 10 cylinder diameters (approximately three times the intrinsic wavelength) is recommended for the simulations to obtain wake structures and hydrodynamic forces that are not strongly restricted by the numerical set‐up. The performances of the periodic and symmetry lateral boundary conditions are compared and discussed. The symmetry boundary condition is recommended for predicting Re_cr, while the periodic boundary condition is recommended for simulating the wake structures above Re_cr. The general conclusions drawn through a circular cylinder are expected to be applicable to other bluff body configurations. Copyright © 2017 John Wiley & Sons, Ltd.     

Flow around a circular cylinder—structure of the near wake shear layer

The separated shear layer in the near wake of a circular cylinder was investigated using a single hot wire probe, with special attention given to the shear layer instability characteristics. Without end plates to force parallel vortex shedding, the critical Reynolds number for the onset of the instability was 740. The present data, together with all previously published data, show that the ratio of the instability frequency f_sl to the vortex shedding frequency f_v varies as Re^0.65, which is in agreement with the Re^0.67 dependence obtained by Prasad and Williamson [1997, J Fluid Mech 333:375–402]. However, the distribution of f_sl/f_v and the spectra of the longitudinal velocity fluctuation (u) suggest that, on either side of Re=5,000, the shear layer exhibits lower and upper subcritical regimes, in support of the observations by Norberg [1987, publication no. 87/2, Chalmers University of Technology, Sweden] and Prasad and Williamson [1997, J Fluid Mech 343:235–265]. The spectra of u provide strong evidence for the occurrence of vortex pairing in wake shear layers, suggesting that the near wake develops in a similar manner to a mixing layer.     

Interference effect of two equal‐sized square cylinders in tandem arrangement: With planar shear flow

Numerical simulations have been performed for flow past two equal‐sized square cylinders in tandem arrangement subjected to incoming planar shear flow. Effect of L/d ratio and the shear parameter has been studied. The range of L/d ratio (ratio of center‐to‐center distance (L) to cylinder width (d)) is varied from 2 to 7 and the non‐dimensional shear parameter (K) is varied from 0.0 to 0.4 in steps of 0.1. For all the cases the Reynolds number (Re) based on centerline velocity and cylinder width is fixed at 100. The results are compared with that of isolated square cylinder with uniform flow. Strouhal number decreases with increasing shear parameter. There are more than one shedding frequency at high shear parameters and L/d ratios. The mean drag coefficient is decreased with shear parameter and lesser than that of the single cylinder. The root mean square (RMS) value of both lift and drag coefficients is higher for the downstream cylinder for all values of shear parameter. With increasing L/d ratio, for both lift and drag, the RMS value increases and then decreases for upstream cylinder, whereas it continuously increases for the downstream cylinder. The stagnation point is moved towards the top leading edge with increasing shear. The critical L/d ratio, which is defined as the distance between two cylinders, beyond which the vortex shedding from the upstream cylinder occurs, decreases with increasing shear parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

Flow past a cylinder: shear layer instability and drag crisis

Flow past a circular cylinder for Re=100 to 10⁷ is studied numerically by solving the unsteady incompressible two‐dimensional Navier–Stokes equations via a stabilized finite element formulation. It is well known that beyond Re ～ 200 the flow develops significant three‐dimensional features. Therefore, two‐dimensional computations are expected to fall well short of predicting the flow accurately at high Re. It is fairly well accepted that the shear layer instability is primarily a two‐dimensional phenomenon. The frequency of the shear layer vortices, from the present computations, agree quite well with the Re^0.67 variation observed by other researchers from experimental measurements. The main objective of this paper is to investigate a possible relationship between the drag crisis (sudden loss of drag at Re ～ 2 × 10⁵) and the instability of the separated shear layer. As Re is increased the transition point of shear layer, beyond which it is unstable, moves upstream. At the critical Reynolds number the transition point is located very close to the point of flow separation. As a result, the shear layer eddies cause mixing of the flow in the boundary layer. This energizes the boundary layer and leads to its reattachment. The delay in flow separation is associated with narrowing of wake, increase in Reynolds shear stress near the shoulder of the cylinder and a significant reduction in the drag and base suction coefficients. The spatial and temporal power spectra for the kinetic energy of the Re=10⁶ flow are computed. As in two‐dimensional isotropic turbulence, E(k) varies as k^?5/3 for wavenumbers higher than energy injection scale and as k^?3 for lower wavenumbers. The present computations suggest that the shear layer vortices play a major role in the transition of boundary layer from laminar to turbulent state. Copyright © 2004 John Wiley & Sons, Ltd.     

Stability of flow past a cylinder: Energy budget of eigenmodes

Global linear stability analysis of the flow past a circular cylinder at the onset of primary wake instability is carried out. The real and imaginary parts of the most unstable eigenmode, responsible for vortex shedding, are very similar but associated with a spatial shift in the vortex structures. This shift results in the convection of vortices that are observed in the unsteady flow, which is actually a consequence of global absolute instability. The kinetic energy density, associated with the most unstable eigenmode, is studied. At the onset of the instability the energy density of the disturbance field is found to be stronger in the far wake compared with the near wake. With increase in Re the region where the disturbance is strong moves upstream closer to the cylinder. However, the maximum value of the kinetic energy density of the disturbance lies outside the recirculation zone even for Re upto 100. A linearized mechanical energy equation for the time evolution of the kinetic energy density of the disturbance is utilized to examine the energy budget of the most unstable eigenmode at various Re. It is found that the most significant contribution to the growth rate of the disturbance arises from the transfer of the energy due to the strain rate of the base flow to the perturbation. The stabilizing effect of the viscous dissipation increases with increase in Re, but saturates for Re beyond ～70. Copyright © 2009 John Wiley & Sons, Ltd.     

Flow past a transversely oscillating square cylinder in free stream at low Reynolds numbers

This paper reports simulation results for free‐stream flow past an oscillating square cylinder at Re=100 and 150, for oscillating‐to‐natural‐shedding frequency ratios of 0.5?f_r?3.0 at a fixed oscillation amplitude of 0.2 of the cylinder width. The transformed governing equations are solved in a non‐inertial frame of reference using the finite volume technique. The ‘lock‐in’ phenomena, where the vortex shedding becomes one with the oscillation frequency, is observed near the natural shedding frequency (f_r≈1). Beyond the synchronization band, downstream recovery of the wake to its stationary (natural) state (frequency) is observed in cross‐stream velocity spectra. At higher forcing frequencies, a phase lag between the immediate and the far wake results in a shear layer having multi‐polar vortices. A ‘Vortex‐switch’ accompanied by a change in the direction of energy transfer is identified at the ‘lock‐in’ boundaries. The variation of aerodynamic forces is noticed to be different in the lock‐in regime. The velocity phase portrait in the far wake revealed a chaotic state of flow at higher excitation though a single (natural) frequency appears in the spectra. Copyright © 2008 John Wiley & Sons, Ltd.     

A numerical investigation of the flow over a pair of identical square cylinders in a tandem arrangement

This paper describes a numerical study of the two‐dimensional and three‐dimensional unsteady flow over two square cylinders arranged in an in‐line configuration for Reynolds numbers from 40 to 1000 and a gap spacing of 4D, where D is the cross‐sectional dimension of the cylinders. The effect of the cylinder spacing, in the range G = 0.3D to 12D, was also studied for selected Reynolds numbers, that is, Re = 130, 150 and 500. An incompressible finite volume code with a collocated grid arrangement was employed to carry out the flow simulations. Instantaneous and time‐averaged and spanwise‐averaged vorticity, pressure, and streamlines are computed and compared for different Reynolds numbers and gap spacings. The time averaged global quantities such as the Strouhal number, the mean and the RMS values of the drag force, the base suction pressure, the lift force and the pressure coefficient are also calculated and compared with the results of a single cylinder. Three major regimes are distinguished according to the normalized gap spacing between cylinders, that is, the single slender‐body regime (G < 0.5), the reattach regime (G < 4) and co‐shedding or binary vortex regime (G ≥4). Hysteresis with different vortex patterns is observed in a certain range of the gap spacings and also for the onset of the vortex shedding. Copyright © 2011 John Wiley & Sons, Ltd.     

Instability of the separated shear layer in flow past a cylinder: Forced excitation

The receptivity of the separated shear layer for Re = 300 flow past a cylinder is investigated by forced excitation via an unsteady inflow. In order to isolate the shear layer instability, a numerical experiment is set up that suppresses the primary wake instability. Computations are carried out for one half of the cylinder, in two dimensions. The flow past half a cylinder with steady inflow is found to be stable for Re = 300. However, an inlet flow with pulsatile perturbations, of amplitude 1% of the mean, results in the excitation of the shear layer mode. The frequency of the perturbation of the inlet flow determines the frequency associated with the shear layer vortices. For a certain range of forced frequencies the recirculation region undergoes a low‐frequency longitudinal contraction and expansion. An attempt is made to relate this instability to a global mode of the wake determined from a linear stability analysis. Interestingly, this phenomenon disappears when the outflow boundary of the computational domain is shifted sufficiently downstream. This study demonstrates the need of carefully investigating the effect of the location of outflow boundaries if the computational results indicate the presence of low‐frequency fluctuations. The effect of Re and amplitude of unsteadiness at the inlet are also presented. All computations have been carried out using a stabilized finite element formulation of the incompressible flow equations. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical study of vortex shedding from a rotating cylinder immersed in a uniform flow field

A numerical study is made of the unsteady two‐dimensional, incompressible flow past an impulsively started translating and rotating circular cylinder. The Reynolds number (Re) and the rotating‐to‐translating speed ratio (α) are two controlled parameters, and the influence of their different combinations on vortex shedding from the cylinder is investigated by the numerical scheme sketched below. Associated with the streamfunction (ψ)–vorticity (ω) formulation of the Navier–Stokes equations, the Poisson equation for ψ is solved by a Fourier/finite‐analytic, separation of variable approach. This approach allows one to attenuate the artificial far‐field boundary, and also yields a global conditioning on the wall vorticity in response to the no‐slip condition. As for the vorticity transport equation, spatial discretization is done by means of finite difference in which the convection terms are handled with the aid of an ENO (essentially non‐oscillatory)‐like data reconstruction process. Finally, the interior vorticity is updated by an explicit, second‐order Runge–Kutta method. Present computations fall into two categories. One with Re=10³ and α≤3; the other with Re=10⁴ and α≤2. Comparisons with other numerical or physical experiments are included. Copyright © 2000 John Wiley & Sons, Ltd.     

Thermal characteristics of the wake shear layers from a slightly heated circular cylinder

In this paper, we investigate the thermal characteristics of wake shear layers generated by a slightly heated circular cylinder. Measurements of the fluctuating temperature were made in the region x/d = 0.6 to x/d = 3 (where x is the downstream distance from the cylinder axis and d is the cylinder diameter) using a single cold-wire probe. The Reynolds number Re was varied in the range 2,600–8,600. For Re = 5,500, simultaneous measurements were made with a rake of 16 cold wires, aligned in the direction of the mean shear, at x/d = 2 and 3. The results indicate that the passive temperature can be an effective marker of various instabilities of the wake shear layers, including the Kelvin–Helmholtz (KH) instability. The temperature data have confirmed the approximate Re ^m dependence of the KH instability frequency (f _KH) with different values of m over different ranges of Re, as reported previously in the literature. However, it is found that this power-law dependence is not exact, and a third-order polynomial dependence appears to fit the data well over the full range of Re. Importantly, it is found that the wake shear-layer instabilities can be grouped into three categories: (1) one with frequencies much smaller than the Bénard–Kármán-vortex shedding frequency, (2) one associated with the vortex shedding and (3) one related to the KH instability. The low-frequency shear-layer instabilities from both sides of the cylinder are in-phase, in contrast to the anti-phase high-frequency KH instabilities. Finally, the observed streamwise decrease in the mean KH frequency provides strong support for the occurrence of vortex pairing in wake shear layers from a circular cylinder, thus implying that both the wake shear layer and a mixing layer develop in similar fashion.     

Effect of self-issuing jets along the span on the near-wake of a square cylinder

The study herein focuses on the vortex shedding characteristics and near-wake vorticity patterns of a square cylinder having self-issuing jets through holes along its span. Three different values of spacing between the consecutive holes λ with respect to the cylinder diameter D, i.e., λ/D = 1.5, 3 and 4 are studied experimentally via Digital Particle Image Velocimetry for the Reynolds number range extending from 200 to 1,000. It has been observed that the three-dimensionality of the wake flow depends on the spacing between the holes and Re number. For sufficiently low Reynolds numbers, the jet flows issuing from the holes yield a non-uniform distribution of mean flow characteristics like the shedding frequency and the formation length of vortices along the span of the cylinder when the spacing between jets along centerline is close to wavelength of the naturally existing three-dimensional wake instability. Additionally, for Re number up to 500, the self-issuing jets emanating from the holes show an indirect interaction with shear layers originating from upper and lower separation lines of the cylinder. However, for higher Re numbers of 750 and 1,000, they directly interact with and modify the vortices forming from the cylinder.     

Role of the shear layer instability in the near wake behavior of two side-by-side circular cylinders

Wakes, and their interaction behind two parallel cylinders lying in a plane perpendicular to the flow, have been investigated experimentally in the sub-critical Reynolds number regime. The experiments were performed in a water channel using laser Doppler velocimetry. The gap between the two cylinders was less than the cylinder diameter, a geometry referred to as strong interaction configuration. In this case the blockage is strong and a gap-jet appears between the cylinders. Two flow regimes of the near wake region have been identified: one below a critical Reynolds number Re _c ]1000;1700[, where the gap jet is stably deflected to one side and the double near-wake becomes asymmetric; the other, above Re _c, where the gap-jet deflection is unstable and a random flopping phenomenon takes place. When Re<Re _c, two different Strouhal numbers are identified, related to the Kármán vortex shedding behind each cylinder. When Re>Re _c, a third frequency appears in the near wake, related to the development of Kelvin-Helmholtz vortices in the separated shear layer of the cylinders [Prasad A, Williamson CHK (1997) J Fluid Mech 333:375]. The observed flopping behavior is attributed to the birth of these Kelvin-Helmholtz instabilities and their intermittent nature. Further downstream, beyond about five cylinder diameters, the random flopping flow phenomena disappear while a slightly asymmetric single wake persists. It is characterized by a Strouhal number St=0.13, a value that one would normally measure behind a single cylinder of twice its diameter.     

Aerodynamic loading on a cylinder behind an airfoil

The interaction between the wake of a rotor blade and a downstream cylinder holds the key to the understanding and control of electronic cooling fan noise. In this paper, the aerodynamic characteristics of a circular cylinder are experimentally studied in the presence of an upstream NACA 4412 airfoil for the cylinder-diameter-based Reynolds numbers of Re_d=2,100–20,000, and the airfoil chord-length-based Reynolds numbers of Re_c=14,700–140,000. Lift and drag fluctuations on the cylinder, and the longitudinal velocity fluctuations of the flow behind the cylinder were measured simultaneously using a load cell and two hot wires, respectively. Data analysis shows that unsteady forces on the cylinder increase significantly in the presence of the airfoil wake. The dependence of the forces on two parameters is investigated, that is, the lateral distance (T) between the airfoil and the cylinder, and the Reynolds number. The forces decline quickly as T increases. For Re_c<60,000, the vortices shed from the upstream airfoil make a major contribution to the unsteady forces on the cylinder compared to the vortex shedding from the cylinder itself. For Re_c>60,000, no vortices are generated from the airfoil, and the fluctuating forces on the cylinder are caused by its own vortex shedding.     

Finite element study of vortex‐induced cross‐flow and in‐line oscillations of a circular cylinder at low Reynolds numbers

Vortex‐induced vibrations of a circular cylinder placed in a uniform flow at Reynolds number 325 are investigated using a stabilized space–time finite element formulation. The Navier–Stokes equations for incompressible fluid flow are solved for a two‐dimensional case along with the equations of motion of the cylinder that is mounted on lightly damped spring supports. The cylinder is allowed to vibrate, both in the in‐line and in the cross‐flow directions. Results of the computations are presented for various values of the structural frequency of the oscillator, including those that are sub and superharmonics of the vortex‐shedding frequency for a stationary cylinder. In most of the cases, the trajectory of the cylinder corresponds to a Lissajou figure of 8. Lock‐in is observed for a range of values of the structural frequency. Over a certain range of structural frequency (F_s), the vortex‐shedding frequency of the oscillating cylinder does not match F_s exactly; there is a slight detuning. This phenomenon is referred to as soft‐lock‐in. Computations show that this detuning disappears when the mass of the cylinder is significantly larger than the mass of the surrounding fluid it displaces. A self‐limiting nature of the oscillator with respect to cross‐flow vibration amplitude is observed. It is believed that the detuning of the vortex‐shedding frequency from the structural frequency is a mechanism of the oscillator to self‐limit its vibration amplitude. The dependence of the unsteady solution on the spatial resolution of the finite element mesh is also investigated. Copyright © 1999 John Wiley & Sons, Ltd.     

Wake transition for surface mounted rectangular cylinder due to incoming shear flow

Three-dimensional unsteady wake characteristics have been investigated numerically in flow past surface mounted finite-height rectangular cylinder using Open Source Field Operation and Manipulation. Effect of impinging shear (shear intensity, K) on transitional characteristics of wake flow has been studied using iso-Q surfaces for Reynolds number (Re) in the range from 150 to 250. Various flow regimes, such as steady flow, symmetric and asymmetric modes of vortex shedding have been identified based on the values of Re and K for different side ratios (SR) of the cylinder. Unsteady wake oscillations have been analyzed using time signal of transverse velocity component in the wake. These signals have been decomposed into different component signals using Hilbert-Huang transformation (HHT). Variation of frequency and energy density with time of the decomposed signals has been presented in the form of Hilbert spectra. Effects of Re, SR and K on wake oscillation frequency have been illustrated in the form of marginal spectra. Time-delay reconstructions and Poincare sections have been examined to study periodic and aperiodic nature of the wake flow. Non-stationarity associated with the wake fluctuation is quantified in terms of degree of stationarity. Symmetric and asymmetric modes have been confirmed using singular value decomposition of the vorticity field and presented using dynamic modes. Growth rate and frequency of the modes corresponding to symmetric shedding are found to be lower than those for asymmetric shedding. In addition, variation in mean drag coefficient has been reported with change in Re and K for each value of SR.