Applied Mathematics and Mechanics (English Edition)

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1 Appl. Math. Mech. -Engl. Ed., 39(9), (2018) Applied Mathematics and Mechanics (English Edition) Direct numerical simulation of turbulent flows through concentric annulus with circumferential oscillation of inner wall Yichen YAO, Chunxiao XU, Weixi HUANG Department of Engineering Mechanics, Tsinghua University, Beijing , China (Received Dec. 24, 2017 / Revised Mar. 11, 2018) Abstract Periodic wall oscillations in the spanwise or circumferential direction can greatly reduce the friction drag in turbulent channel and pipe flows. In a concentric annulus, the constant rotation of the inner cylinder can intensify turbulence fluctuations and enhance skin friction due to centrifugal instabilities. In the present study, the effects of the periodic oscillation of the inner wall on turbulent flows through concentric annulus are investigated by the direct numerical simulation (DNS). The radius ratio of the inner to the outer cylinders is 0.1, and the Reynolds number is based on the bulk mean velocity U m and the half annulus gap H. The influence of oscillation period is considered. It is found that for short-period oscillations, the Stokes layer formed by the circumferential wall movement can effectively inhibit the near-wall coherent motions and lead to skin friction reduction, while for long-period oscillations, the centrifugal instability has enough time to develop and generate new vortices, resulting in the enhancement of turbulence intensity and skin friction. Key words turbulence, drag reduction, circumferential oscillation, Taylor vortex Chinese Library Classification O Mathematics Subject Classification 76F75, 76F70, 76U05 1 Introduction Friction drag reduction in wall turbulence is of great significance for many engineering applications. Considering the global trend of energy shortage and pollution aggravation, much attention has been focused on the control strategies to reduce the turbulence friction drag. In the 1980s, Bradshaw and Pontikos [1] made wind tunnel experiments of swept wing and found that the sudden spanwise pressure gradient could generate a transient drop in turbulent friction, and near-wall flow would gradually recover and realign in the new oblique direction. Inspired by this observation, Jung et al. [2] and Akhavan et al. [3] carried out a DNS of a channel flow and achieved sustained attenuation of turbulence intensities when high-frequency spanwise oscillation was applied to the wall. Laadhari et al. [4] also confirmed the performance of wall oscillation control in the boundary layer experiments. This drag reduction control scheme of Citation: YAO, Y. C., XU, C. X., and HUANG, W. X. Direct numerical simulation of turbulent flows through concentric annulus with circumferential oscillation of inner wall. Applied Mathematics and Mechanics (English Edition), 39(9), (2018) Corresponding author, xucx@tsinghua.edu.cn Project supported by the National Natural Science Foundation of China (No ) c Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

2 1268 Yichen YAO, Chunxiao XU, and Weixi HUANG lateral wall movement has several advantages. Besides the considerable drag reduction rate, it is easy to be implemented in engineering applications, and no feedback information is needed. For this lateral oscillation control, the drag reduction effectiveness relies on the oscillation parameters. In the channel flow, it is found that with increasing oscillation amplitude, drag reduction increases monotonously, and an optimal control period T is obtained (the superscript + denotes the normalization by wall units). According to the numerical simulation results, a 45% drag reduction and a 7% net energy saving could be achieved at the best control parameters [5]. Based on the influence range of the Stokes layer, Choi et al. [6] obtained a combined fitting function which correlated well with the drag reduction curve. Quadrio et al. [7] and Quadrio and Ricco [8] focused on the sinusoidal spanwise travelling wave in the streamwise direction and constructed the relationship between the drag reduction performance and the near-wall turbulence convective velocity. In explanation of this oscillation control strategy, Choi et al. [9] and Choi [10] attributed the drag reduction mechanism to the interaction of Stokes layer and the quasi-streamwise vortices. The production of negative spanwise vorticity during the oscillation cycle reduces the mean velocity gradient in the near-wall region. While Choi et al. [6] proposed that the vortical structure lies at a higher wall-normal position than streaks, and the continuous wall oscillation provides a relative phase shift of these two structures, thus the self-sustain mechanism of the near-wall turbulence can be suppressed. Despite the substantial investigations of the drag reduction performance achieved by the lateral wall oscillation, researchers are mainly concentrated on flat wall turbulence or flow inside the circular pipe. Turbulent flows between concentric annular pipes were seldom considered. It is known that in the concentric annulus, the constant rotation of the inner cylinder can enhance the near-wall ejection and sweep events, thus causing large friction increase [11 13]. Thus, the performance of the periodic oscillation of the inner cylinder on turbulence suppression and friction drag reduction needs to be evaluated in the concentric annulus. In the present study, the main purpose is to build the relationship between the near-wall flow pattern and the imposed wall movement and evaluate the overall drag reduction behavior. Particular attention is devoted to the destabilizing effect due to the azimuthal curvature of the flow geometry, in which the emergence of new vortices can drastically change the near-wall flow structures and cause friction drag enhancement. 2 Problem formulation and numerical method The DNS is performed to turbulent flows driven by a given axial pressure gradient in a concentric annulus with the inner cylinder periodically oscillating in the circumferential direction. The Navier-Stokes equations for the incompressible Newtonian fluids are solved directly by pseudo-spectral methods. Periodic boundary conditions are applied in the axial and circumferential directions, and the non-slip condition is imposed on the walls. The Chebyshev-collocation method in the radial direction and Fourier-Galerkin method in the streamwise and circumferential directions are used for spatial discretization, and the third-order time-splitting method is adopted for time advancement [14]. The flow geometry is shown in Fig.1. The length of the computational domain in the streamwise direction is 2πH. The grid numbers are in the streamwise, radial, and circumferential directions, respectively, and the grid sizes in the three directions are respectively x + = 4.14, r + min / max = , and r θ+ min /max = for the uncontrolled case. The amplitude of the circumferential velocity is equal to the bulk mean velocity, and the radius ratio is 0.1 between the inner and outer cylinder. No feedback is required for the control, and the imposed periodic circumferential velocity on the inner cylinder is described as u θ,inner = U m sin(2πt/t). The Reynolds number based on the bulk mean velocity U m and half annulus gap H is A series of oscillation periods, i.e., T = 1, 2,, 15H/U m, are considered to reveal the relationship between the forcing period and the drag reduction behavior. The flow

3 Direct numerical simulation of turbulent flows through concentric annulus 1269 quantities are normalized by U m and H, except for those with the superscript +, which are normalized by the kinematic viscosity ν and the friction velocity u τ of the uncontrolled flow. Flow 3 Results and discussion Fig. 1 Flow geometry 3.1 Basic statistics The drag reduction rate (D R ) is defined as (τ no τ c )/τ no, where τ no denotes the wall shear stress of the inner cylinder in the no control case, and τ c is that with circumferential wall oscillation control. Figure 2 shows D R at different oscillation periods. D R increases with the oscillation period when T < 8 and reaches an optimal value of 28% around T = 8, which is equivalent to T + = 116. After that, D R drops abruptly and becomes negative when T > 9. This phenomenon has not been observed in the previous studies of the flat-wall oscillation control. The friction Reynolds number and the corresponding D R at T = 8 and T = 10 are listed in Table 1, together with those for the static and constant rotating wall cases for comparison. Fig. 2 Mean drag reduction rate as a function of oscillation period Table 1 Drag reduction rate (D R/%) behavior of different control cases Condition Re τ,inner D R /% Static Rotating T = T = Figure 3 shows the time evolution of the instantaneous drag reduction rate. After the initial settling-down periods, the drag is almost time independent at the optimal period T = 8. However, at the longer oscillation period T = 10, the phase-related drag fluctuation is observed. The reason for this behavior will be analyzed in the next subsections.

4 1270 Yichen YAO, Chunxiao XU, and Weixi HUANG Fig. 3 Time evolution of skin friction at the inner cylinder wall (color online) The response of near-wall turbulence to the wall oscillation can be quantitatively reflected in turbulence statistics. For the mean streamwise velocity profile in Fig. 4, an obvious extension of the buffer layer and upshift of the log layer can be observed for the T = 8 case, in accordance with the decreasing friction drag. On the contrary, for the constant rotating case and T = 10 oscillation case, the buffer layer becomes thinner, and the log layer moves downward. Fig. 4 Mean streamwise velocity profile normalized by the uncontrolled case (color online) Figure 5(a) shows Reynolds stress distribution u r u x normalized by wall shear stress of the uncontrolled case. For the optimal control period T = 8, the peak position of the Reynolds stresses shift from y + = 23 to y + = 30, indicating that the buffer layer becomes thick. On the contrary, for the T = 10 case, the near-wall Reynolds stresses increase by 20%, and its peak position moves slightly towards the wall. Mean square values of velocity fluctuations in the streamwise, radial, and circumferential directions are shown in Figs. 5(b) 5(d). For the T = 8 case, the second-order statistics in the near-wall region decrease evidently, indicating that turbulence is successfully depressed. While for the T = 10 and constant rotating cases, although the streamwise velocity fluctuations are attenuated, the velocity fluctuations in the radial and circumferential directions are significantly increased. In the laminar channel flows, the governing equation for the spanwise velocity caused by periodic wall oscillation is decoupled from that of the streamwise velocity component, and its analytical solution is equivalent to the Stokes second problem. The velocity envelope decays exponentially with the wall-normal distance. For the turbulent flows, the modulation of spanwise velocity from turbulence is limited, and the velocity profile resembles that of the Stokes second problem. In the concentric annulus flow under discussion, the circumferential velocity is decoupled from the other two velocity components, and the momentum equation in the

5 Direct numerical simulation of turbulent flows through concentric annulus 1271 circumferential direction satisfies u θ t ( 2 u θ = ν r u θ r r u ) θ r 2. (1) Fig. 5 Second-order statistics of velocity fluctuations (color online) The analytical solution of the above equation is expressed as (( Y 1 (R 2 ) u θ (r, t) = Im J 1 (R 1 )Y 1 (R 2 ) J 1 (R 2 )Y 1 (R 1 ) J 1(F(r)) + J 1 (R 2 ) Y 1 (R 1 )J 1 (R 2 ) Y 1 (R 2 )J 1 (R 1 ) Y 1(F(r)) ) exp ( 2πit )) U m, (2) T where F(r) = r(1 i) π/(νt), J 1 and Y 1 are respectively the first-order Bessel function of the first kind and the first-order Bessel function of the second kind, R 1 and R 2 are the radii of the inner and outer cylinders. For the T = 8 case with oscillation period below the critical value, the phase-averaged circumferential velocity approximately coincides with the corresponding laminar solution (see Fig. 6(a)). At this state, the influence range of the Stokes layer is mainly below the buffer region. The velocity streaks mainly lie below the buffer layer, while the quasi-streamwise vortices emerge at relatively higher positions, and thus the continuous circumferential phase shift between these two types of structure leads to the disruption of the near-wall self-sustaining process. With control periods beyond T = 9, there presents an obvious change in the mean circumferential velocity profile compared with the laminar case (see Fig. 6(b)). The wall-normal distribution of circumferential velocity is more flattened compared with the laminar case, indicating the enhanced momentum exchanges along the radial direction.

6 1272 Yichen YAO, Chunxiao XU, and Weixi HUANG (a) = 8 (b) = 10 Fig. 6 Comparison of the analytical solution of Stokes second problem (Stokes BL) and the mean circumferential velocity profile obtained by the present DNS (color online) 3.2 Vortices induced by centrifugal instability Figure 7 shows the phase-averaged drag reduction rate in a single control period for T = 10. The friction drag reaches the maximum value when the wall velocity is zero and reaches the minimum value at the phase of largest wall speed. The strong phase dependency is mainly attributed to the periodic emergence and disappearance of vortices induced by centrifugal instability. /% Fig. 7 Circumferential velocity of the wall and the corresponding drag reduction rate in one control period for the T = 10 case Figure 8 shows the instantaneous circumferential velocity in a meridian plane at phase angle 0 and 2π/5. Vortical structures are most pronounced at the phase of zero wall velocity, and they are periodically distributed along the axial direction in the near-wall region as shown in Fig. 8(a). As for the accumulation of wall displacement in the same direction in the half control period, near-wall structures have enough time to adapt to its natural equilibrium state and form the vortical structures. Under this scenario, the vortices induced by centrifugal instability become the dominant structures in the near-wall region. These vortices eject low-speed fluids away from the wall and entrain high-speed fluids towards the wall, leading to the enhanced momentum transport in the radial direction. At the phase of 2π/5, the inner cylinder is rotating towards the opposite circumferential direction, and the vortical structures formed previously are destroyed while the new vortices are not fully developed, as shown in Fig. 8(b), the vortical structures become weak and the wall friction reaches its minimum value.

Direct numerical simulation of turbulent flows through concentric annulus 1273 π Fig. 8 Instantaneous circumferential velocity for the T = 10 case (color online) 3.

point. Joint probability density function (PDF) of velocity fluctuations at y + 10 for T = 8, T = 10, and the uncontrolled cases is shown in Fig.9.

In the figure, the horizontal axis is the streamwise fluctuation u x, and the vertical axis is the radial fluctuation u r.

7 Direct numerical simulation of turbulent flows through concentric annulus 1273 π Fig. 8 Instantaneous circumferential velocity for the T = 10 case (color online) 3.3 Contribution to turbulence production from instability induced vortices The flow structure and turbulence statistics present clear difference below and above the centrifugal instability critical point. Joint probability density function (PDF) of velocity fluctuations at y + 10 for T = 8, T = 10, and the uncontrolled cases is shown in Fig.9. The contour lines in the figure respectively indicate the 10% to 90% of the maximum probability with the interval of 20%. In the figure, the horizontal axis is the streamwise fluctuation u x, and the vertical axis is the radial fluctuation u r. Thus the second and fourth quadrants of this phase plane correspond to turbulent ejection and sweep events respectively. For the T = 8 case, the contour converges towards the origin point, indicating that turbulence is effectively suppressed. For the T = 10 case, the velocity fluctuation decreases in the streamwise direction, and is highly intensified in the radial direction due to the instability induced vortices. Fig. 9 Joint PDF of streamwise and radial velocity fluctuations (color online) As discussed above, the velocity fluctuation in the radial direction can promote the near-wall turbulence intensity, and we resort to the production term of turbulent kinetic for further investigation. In the cylindrical coordinates, the turbulent production term of the circumferential oscillation control under discussion is expressed as ( uθ P k = r + u ) θ u r r u θ u x r u r u x. (3) Figure 10 shows the phase-averaged production term for T = 10, in which Fig. 10(a) presents u ru θ ( u θ r + u θ /r), and Fig. 10(b) presents u ru x ux r. Obviously, both terms present a high dependency on oscillation phase. The term related with stress u r u θ reaches its maximum at phase 0.4T and 0.9T, which is much larger than that of other phases. At this particular phase, the contribution from the term related with stress u ru θ approaches that of the term

8 1274 Yichen YAO, Chunxiao XU, and Weixi HUANG related with stress u r u x in magnitude, which is consistent with the results of PDF analysis. The same analysis of turbulent production term for T = 8 presents a different scenario as shown in Fig. 11. At either phase angle 0 or π/2, the turbulent production term is one magnitude lower than that of T = 10. Moreover, the term with u r u θ becomes less significant compared with the u ru x term. Fig. 10 Production term of turbulent kinetic energy for the T = 10 case (color online) Fig. 11 Production term of turbulent kinetic energy for the T = 8 case (color online) With regard to the influence of wall oscillation on the instantaneous velocity field, Fig. 12 displays the snapshot of streamwise velocity fluctuation at y + 10, where T = 8 and T = 10 cases are at the phase angle 0. For the uncontrolled case, velocity streaks are mainly in the streamwise direction. However, with the optimal oscillation period of T = 8, it is evident that both the high-speed streaks and the low-speed counterparts are attenuated. On the contrary, for the T = 10 case, streak intensity is strengthened and the inter-streak distance becomes small. As is analyzed by Touber and Leschziner [15], the streak angle follows closely with the phase-averaged shear strain vector at y + 10, and the sinuous wall-motion results in periodic disruption and reformation of streaks. Inhibited streaks further lead to the weakening of the streamwise vortices, thus the near-wall turbulent structures are effectively attenuated. 4 Conclusions In this paper, the DNS is performed to turbulent flows in a concentric annulus with periodic circumferential oscillations of the inner cylinder. The relationship between oscillation periods

Direct numerical simulation of turbulent flows through concentric annulus 1275 Fig. 12 Instantaneous streamwise velocity fluctuation at y + = 10 (color online) and drag reduction rate is investigated.

For the drag reduction cases, both the streaks and the Reynolds stresses are significantly weakened due to the shielding effect of the transverse Stokes layer.

This is due to the emergence of vortices induced by the centrifugal instability. Therefore, turbulence intensity is enhanced near the wall and skin friction is increased.

9 Direct numerical simulation of turbulent flows through concentric annulus 1275 Fig. 12 Instantaneous streamwise velocity fluctuation at y + = 10 (color online) and drag reduction rate is investigated. It is found that at small oscillation periods, the drag reduction rate increases with the oscillation period. The maximum drag reduction of 28% is obtained at T = 8. For the drag reduction cases, both the streaks and the Reynolds stresses are significantly weakened due to the shielding effect of the transverse Stokes layer. When the oscillation period is beyond the critical value T 9, the drag reduction rate decreases abruptly and drops to negative values. This is due to the emergence of vortices induced by the centrifugal instability. Therefore, turbulence intensity is enhanced near the wall and skin friction is increased. The instability-induced vortices are most evident at the phase of maximum wall displacement (with zero wall velocity). The vortices enhance the wall-normal momentum transport, giving rise to the increase in the production term of the turbulent kinetic energy. References [1] BRADSHAW, P. and PONTIKOS, N. S. Measurements in the turbulent boundary layer on an infinite swept wing. Journal of Fluid Mechanics, 159, (1985) [2] JUNG, W. J., MANGIAVACCHI, N., and AKHAVAN, R. Suppression of turbulence in wallbounded flows by high-frequency spanwise oscillations. Physics of Fluids A: Fluid Dynamics, 4(8), (1992) [3] AKHAVAN, R., JUNG, W. J., and MANGIAVACCHI, N. Turbulence control in wall-bounded flows by spanwise oscillations. Advances in Turbulence IV, Springer, Netherlands, (1993)

10 1276 Yichen YAO, Chunxiao XU, and Weixi HUANG [4] LAADHARI, F., SKANDAJI, L., and MOREL, R. Turbulence reduction in a boundary layer by a local spanwise oscillating surface. Physics of Fluids, 6(10), (1994) [5] QUADRIO, M. and RICCO, P. Critical assessment of turbulent drag reduction through spanwise wall oscillations. Journal of Fluid Mechanics, 521, (2004) [6] CHOI, J. I., XU, C. X., and SUNG, H. J. Drag reduction by spanwise wall oscillation in wallbounded turbulent flows. AIAA Journal, 40(5), (2002) [7] QUADRIO, M., RICCO, P., and VIOTTI, C. Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction. Journal of Fluid Mechanics, 627, (2009) [8] QUADRIO, M. and RICCO, P. The laminar generalized Stokes layer and turbulent drag reduction. Journal of Fluid Mechanics, 667, (2011) [9] CHOI, K. S., DEBISSCHOP, J. R., and CLAYTON, B. R. Turbulent boundary-layer control by means of spanwise-wall oscillation. AIAA Journal, 36(7), (1998) [10] CHOI, K. S. Near-wall structure of turbulent boundary layer with spanwise-wall oscillation. Physics of Fluids, 14(7), (2002) [11] CHUNG, S. Y., RHEE, G. H., and SUNG, H. J. Direct numerical simulation of turbulent concentric annular pipe flow: part 1, flow field. International Journal of Heat and Fluid Flow, 23(4), (2002) [12] CHUNG, S. Y. and SUNG, H. J. Large-eddy simulation of turbulent flow in a concentric annulus with rotation of an inner cylinder. International Journal of Heat and Fluid Flow, 26(2), (2005) [13] JUNG, S. Y. and SUNG, H. J. Characterization of the three-dimensional turbulent boundary layer in a concentric annulus with a rotating inner cylinder. Physics of Fluids, 18(11), (2006) [14] KARNIADAKIS, G. E., ISRAELI, M., and ORSZAG, S. A. High-order splitting methods for the incompressible Navier-Stokes equations. Journal of Computational Physics, 97(2), (1991) [15] TOUBER, E. and LESCHZINER, M. A. Near-wall streak modification by spanwise oscillatory wall motion and drag-reduction mechanisms. Journal of Fluid Mechanics, 693, (2012)

Drag reduction in wall-bounded turbulence via a transverse travelling wave

Drag reduction in wall-bounded turbulence via a transverse travelling wave Downloaded from https://www.cambridge.org/core. Brown University Library, on 27 Mar 218 at 19:12:8, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/1.117/s2211217613