As a bluff-body is immersed in the shear flow,a negative lift force may be generated, similar to the stall phenomenon encountered in airfoils at high attack angles,which is generally undesired from the practical point of view. Therefore,the investigations on underlay mechanism are also necessary owing to the theoretical and practical importance^{[1, 2, 3]}.

The simplest case is a uniform-shear flow past an isolated circular cylinder,with a linear distribution of the velocity components in the lateral direction and an infinite lateral width of the flow domain. Part of the early studies on shear flow was summarized by Jordon and Fromm^[4],which was mainly focusing on the shift of the front stagnation point^[5] and the motion of the separation points^[6]. Recently,Lei et al.^[7] and Kang^[8] paid more attention to the effect of Reynolds number and shear rate on the vortex-shedding frequency,the magnitude and direction of the mean lift,the magnitude of the mean drag,and so on. Meanwhile,experimental researches on shear flow were reported by Kiya et al.^[9],Kwon et al.^[10],Hayashi et al.^[11],and Sumner and Akosile^[12]. However,it is clear that there still remain some uncertainties regarding the vortex shedding and hydrodynamic forces acting on a circular cylinder in the shear flow, and more investigations on the uniform-shear flow are necessary.

In this study,the characteristics of a uniform-shear flow over a circular cylinder are investi- gated numerically. The stream function-vorticity equations in the exponential-polar coordinates are solved by using the alternative-direction implicit (ADI) algorithm and fast Fourier trans- form (FFT) one. Then,the distributions of pressure and drag and lift on the surface of cylinder are directly obtained from the corresponding equations derived in the paper. It is revealed from the calculations that the closed shape and the position of the diagram of lift-drag phase can not only denote the corresponding variations of the lift and drag over a complete period,but also imply the time history of the fluctuating pressures on the cylinder surface and the detail information of the flow pattern. Therefore,the lift-drag phase diagram only mentioned simply by Kang is used in present study for further understanding the wake dynamics and hydrody- namic forces in the uniform-shear flow over a circular cylinder. The effects of the shear rate on the lift-drag phase diagram,subsequently related with the vortex shedding and the pressure distribution,are discussed in detail to examine the corresponding underlying mechanism. 2 Numerical method

uniform plan shear flow with a linear velocity profile u = u_c + Gy^[7] over a cylinder in two-dimensional approach has been investigated numerically,where y is a coordinate in the lateral direction with y =0 at the center of the cylinder,uc and G are the streamwise velocity at the center-line y = 0,and the lateral velocity gradient,respectively. Only the case of shear rate (K > 0) is discussed in the paper,where the shear rate K is defined as K = 2Ga/u_c,a is the cylinder radius.

The stream function-vorticity equations in the exponential-polar coordinates (ξ,η),r = e^2πξ, θ = 2πη for the incompressible fluid are written as follows:

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Based on the analytic solutions of non-viscous flow with slip boundary conditions^[6],the initial flow field at t=0 is described by the following equations:

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At t > 0,the no-slip boundary condition is used instead of the slip boundary condition. On the cylinder surface ξ = 0,we have ψ = 0,Ω = −1/H∂₂ψ/∂ξ₂ ,and in ξ = ξ_∞,we have

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The momentum equation along ξ-direction is

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The shear stress C^θ_τ can be expressed as

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The total hydrodynamic force obtained by integrating the force distribution function along the cylinder surface consists of drag force C_d and lift force C_l denoting the force components in the streamwise and the normal directions,respectively. We can then write

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Equation (1) is solved by using the ADI algorithm,and Eq. (2) is integrated by means of a FFT one. More details can be found in Refs. ^{[3, 13, 14]}. Numerical results are obtained at Re=150 with the numerical step size Δξ = 0.004,Δη = 0.002,and Δt = 0.005. 3 Results and discussion 3.1 Uniform flow over cylinder

Numerical results for cylinder wake in the uniform flow (Re = 150) at four typical moments of a shedding period are exhibited by the contours of vorticity in Fig. 1,where the red vortexes are negative and blue are positive. For convenience,the time at t=0,T /4,T /2,and 3T /4 are denoted by T_D,T_A,T_B and T_C,respectively,where T denotes one period of vortex shedding.

At T_D,the vortex generated from the shear layer of lower side becomes a dominant vortex in the near-wake flow shown in Fig. 1(a),which T_Akes the upper shear layer into itself,and entrains the upwind flow. The upper vortex begins to generate. Both the upper and lower separation points are at the undeflected positions.

As lower vortex moves downstream with the decreasing strength,the upper shear layer is rolled up gradually. Till T_A,when the roll-up of the upper shear layer has been clearly visualized in Fig. 1(b),the interaction of the upper and lower shed vortices becomes so strong that the near weak flow is affected equally by the two shed vortices. Both the lower and upper separation points reach its dead-end.

Fig. 1. Numerical results for cylinder wake at typical moments with K=0 and Re=150

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For further evolutions,it is the upper shed vortex that becomes the dominant one in the near-wake flow at T_B,while the portion of the lower shed vortex near the separation point begins to roll up as indicated in Fig. 1(c),where the flow field is symmetric about θ = 0◦ with their counterparts in Fig. 1(a). Both the upper and lower separation points are at the undeflected positions again. However,the upper separation point tends to move downstream and the lower separation point tends to move upstream.

Till T_C shown in Fig. 1(d),the influence of the lower vortex prevails over the upper vortex, the flow field is symmetric about θ = 0◦ at T_C with that at T_A. Both the upper and lower separation points reach its dead-end. Finally,at T_D,shown in Fig. 1(a) again,vortex shed has completed for one entire periodic.

As mentioned above,if T_A is T_Aken as the start of a shedding cycle,i.e.,t=0,then in the first half cycle,the upper vortex is prevalent in the near-wake,and then the lower vortex is prevalent in the second half cycle.

It is generally known^{[15, 16]} that the pressure is much larger than the shear stress. Therefore, it is essential to know the distribution of pressure on the surface of cylinder. The distribution of pressure coefficient on the cylinder surface can be obtained from Eq. (6) and shown in Fig. 2 at different time instants in one cycle. At T_A and T_C,the vortices are not strong behind the cylinder,so that the profile of pressure on the lower side of the cylinder is mirrored to the upper side. Therefore,the drag is small and the lift is zero.

Since vortex shed increases the velocity of fluid near the surface and entrains the upwind flow,the pressure decreases along the upwind surface from the front stagnation point on the shedding side,and its decreasing amplitude depends on the vortex strength. At T_B and T_D, when a single vortex dominates the near wake,the profile of pressure moves down on the side,where the shedding vortex dominates the near wake,and moves up on the opposite side compared with the curves at T_A and T_C as shown in Fig. 2. Therefore,the lift tends to the maximum value at T_B and T_D. So does the drag.

Fig. 2. Pressure coefficient distributions on cylinder surface for K=0

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The lift and drag force behaviors induced by the vortex shedding may be performed more clearly in the lift-drag phase diagram by plotting C_l as a function of C_d,which has a closed shape corresponding to the Lissajou figure as shown in Fig. 3.

Fig. 3. Lift-drag phase diagram for K=0

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The phase diagram of lift-drag is symmetric about the line C_l = 0. On the curve ABC, corresponding with the upper loop,the lift coefficient is positive,owing to the upper vortex prevailed. Whereas on the CDA curve,corresponding to the lower loop owing to the prevailed lower vortex,the coefficient of lift is negative. 3.2 Shear flow over cylinder

Based on the Ref. ^[6],the adverse flow is generated on the side whose flow speed is slower while the value of shear rate K is larger. In this condition,the number of separation point changes from 1 into 2 on the side which means the change of flow field character. Therefore, the cases with low shear rate (K<0.5) are discussed in this paper which the flow directions on both sides of cylinder are the same.

Numerical results for cylinder wake in a shear flow with Re=150 and K=0.46 at four typical moments of a shedding period are exhibited by the contours of vorticity in Fig. 4.

Fig. 4. Numerical results for cylinder wake at typical moments for shear flow with Re=150 and K=0.46

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It is noteworthy that the asymmetry of the free stream causes a shift of the front stagnation point towards the high velocity side,and results in the decline of vortex streets towards the lower velocity side.

At T_D,the vortex generated from the lower shear layer becomes the dominant one and the upper vortex begins to generate. The front stagnation point shifts to get its minimum. At T_A, the roll-up of the upper shear layer has been clearly visualized. The stagnation point gets its equilibrium positions and moves downstream. At T_B,the upper vortex becomes the dominant one,while the lower shear layer begins to roll up. The stagnation point gets its maximum. Since the negative background vorticity in the free-stream increases the strength of upper shear layer and vortices,and decreases that on the lower side. Then,the strength of upper vortex at T_B is larger than that of lower vortex at T_D. At T_C,the roll-up of the lower shear layer has been clearly visualized. The stagnation point gets its equilibrium positions and moves upstream. Finally,at T_D,vortex shed has completed for one entire periodic.

As shown in Fig. 5,the shift of the stagnation point causes an increased amount of fluid to pass over the bottom of the cylinder and thus shifts the positive pressure area to the same side. Meanwhile,the strengthened (or weakened) vortices on the upper (or lower) side can increase the pressure on the surface of upper windward and leeward.

Fig. 5. Pressure coefficient distributions on cylinder surface for different time instants and shear rates

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The increased pressure on the upper surface causes the lift to be negative,exerted from the upper side to the lower side,whereas the increased pressure on the leeward surface can reduce the drag. Therefore,the position of A (or C) in the lift-drag phase diagram will move down and left as the increased shear rate,as shown in Fig. 6.

Fig. 6. Variations of lift-drag diagram phase with shear rate K

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Due to the background vorticity in the shear flow,the effect of the vortex shed on the curve ABC is strengthened,while the effect on the curve CDA is weakened. Therefore,the area of the curve CDA shrinks considerably compared with the curve ABC.

It is also seen in Fig. 6 that the drag on the curve CDA is smaller completely than that on the curve ABC as long as the shear rate K is large enough,which implies the drag at A (or C) is minimum on the curve ABC,but is maximum on the curve CDA,which characteristic is distinctly different with that for K = 0.

To elucidate the drag reduction in the curve CDA further,the pressure coefficient distribu- tions on the cylinder surface at different time instants in one cycle for shear flow with K = 0.46 are depicted in Fig. 7,where the mirrored symmetry in the case of K = 0 has been broken. The obvious changes of pressure distributions appear mainly in the leeward surface of the cylinder, which affects the drag directly since the drag depends on the difference of pressure distribution between the windward and leeward.

Fig. 7. Pressure coefficient distributions on cylinder surface in shear flow with K = 0.46

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Based on the calculated results,the base pressure defined as the pressure integration over the leeward surface in the paper is dominated by the distribution of pressure on the upper leeward surface,where the shedding vortices are strengthened by the background vorticity. Therefore, the base pressure at T_D is higher than that at T_A (or T_C),whereas the base pressure at T_B is the lowest. The higher the base pressure is,the lower the drag is. Hence,the drag on the curve CDA is smaller than that on the curve ABC. 4 Conclusions

The uniform-shear flow past a circular cylinder has been investigated numerically in the exponential-polar coordinates. The results can be summarized as follows:

(i) Four typical instants,T_A ξ T_D have been detected from instantaneous flow fields in one shedding cycle. The vortex shed from one side becomes the prevailing one,and the onset of the vortex shedding on the other side just occurs at T_B and T_D. Whereas the vortices shed from the upper and lower sides affect the weak flow equally at T_A and T_C.

(ii) The pressure distributions on the cylinder surface,affecting the drag and lift force strongly,depends directly on the pattern of the weak flow. The vortex shedding contributes to the pressure decrease on the shedding side due to the entraining fluid. In addition,in the shear flow,the background vorticity and the stagnant point shift induced also affect the pressure distribution considerably.

(iii) In the shear flow,the symmetry of the lift-drag phase diagram is broken. The position of the crossed point of the diagram will move down and left as the increased shear rate,and the area of the curve CDA shrinks considerably compared with the curve ABC. Furthermore, the drag on the curve CDA is smaller completely than that on the curve ABC as long as the shear rate K is large enough.