Minimum power consumption for drag reduction on a circular cylinder by tangential surface motion.(English)Zbl 1284.76129

Summary: We investigate the effect of a prescribed tangential velocity on the drag force on a circular cylinder in a spanwise uniform cross flow. Using a combination of theoretical and numerical techniques we make an attempt at determining the optimal tangential velocity profiles which will reduce the drag force acting on the cylindrical body while minimizing the net power consumption characterized through a non-dimensional power loss coefficient ($$C_{PL}$$). A striking conclusion of our analysis is that the tangential velocity associated with the potential flow, which completely suppresses the drag force, is not optimal for both small and large, but finite Reynolds number. When inertial effects are negligible ($$Re\ll 1$$), theoretical analysis based on two-dimensional Oseen equations gives us the optimal tangential velocity profile which leads to energetically efficient drag reduction. Furthermore, in the limit of zero Reynolds number ($$Re\to 0$$), minimum power loss is achieved for a tangential velocity profile corresponding to a shear-free perfect slip boundary. At finite $$Re$$, results from numerical simulations indicate that perfect slip is not optimum and a further reduction in drag can be achieved for reduced power consumption. A gradual increase in the strength of a tangential velocity which involves only the first reflectionally symmetric mode leads to a monotonic reduction in drag and eventual thrust production. Simulations reveal the existence of an optimal strength for which the power consumption attains a minima. At a Reynolds number of $$100$$, minimum value of the power loss coefficient ($$C_{PL}=0.37$$) is obtained when the maximum in tangential surface velocity is about one and a half times the free stream uniform velocity corresponding to a percentage drag reduction of approximately 77%; $$C_{PL}=0.42$$ and $$0.50$$ for perfect slip and potential flow cases, respectively. Our results suggest that potential flow tangential velocity enables energetically efficient propulsion at all Reynolds numbers but optimal drag reduction only for $$Re\to \infty$$. The two-dimensional strategy of reducing drag while minimizing net power consumption is shown to be effective in three dimensions via numerical simulation of flow past an infinite circular cylinder at a Reynolds number of $$300$$. Finally a strategy of reducing drag, suitable for practical implementation and amenable to experimental testing, through piecewise constant tangential velocities distributed along the cylinder periphery is proposed and analysed.

MSC:

 76D55 Flow control and optimization for incompressible viscous fluids

Keywords:

drag reduction; flow control; propulsion
Full Text:

References:

 [1] DOI: 10.1063/1.2033624 · Zbl 1187.76044 [2] DOI: 10.1016/j.jcp.2008.04.034 · Zbl 1388.76073 [3] DOI: 10.1103/PhysRevLett.100.204501 [4] DOI: 10.2514/3.13190 [5] DOI: 10.1017/S0022112070001428 · Zbl 0193.26202 [6] DOI: 10.1103/PhysRevLett.96.134501 [7] DOI: 10.1063/1.2236305 [8] DOI: 10.1146/annurev.fluid.39.050905.110149 · Zbl 1136.76022 [9] DOI: 10.1063/1.3147935 · Zbl 1183.76381 [10] An Introduction to Fluid Dynamics (1967) · Zbl 0152.44402 [11] DOI: 10.1063/1.1491251 · Zbl 1185.76086 [12] DOI: 10.1017/jfm.2011.172 · Zbl 1241.76043 [13] DOI: 10.1017/S0022112096002777 · Zbl 0882.76028 [14] DOI: 10.2514/2.1017 [15] DOI: 10.1017/S0022112063000665 · Zbl 0112.19003 [16] DOI: 10.1006/jcph.2002.7145 · Zbl 1021.76040 [17] DOI: 10.1017/jfm.2011.134 · Zbl 1241.76341 [18] DOI: 10.1006/jfls.1997.0098 [19] DOI: 10.1016/S0045-7930(98)00002-4 · Zbl 0964.76066 [20] J. Fluid Mech. 476 pp 303– (2003) [21] Handbook of Mathematical Functions (1968) [22] DOI: 10.1063/1.868515 · Zbl 1027.76519 [23] DOI: 10.1006/jcph.1996.0065 · Zbl 0849.76064 [24] DOI: 10.1063/1.869912 · Zbl 1147.76329 [25] DOI: 10.1006/jcph.2001.6882 · Zbl 1168.74470 [26] DOI: 10.1063/1.2171193 · Zbl 1185.76556 [27] DOI: 10.1063/1.2756578 · Zbl 1182.76862 [28] DOI: 10.1146/annurev.fluid.32.1.659 · Zbl 0989.76082 [29] DOI: 10.1063/1.2786373 · Zbl 1182.76857 [30] DOI: 10.1006/jcph.1997.5859 · Zbl 0908.76064 [31] DOI: 10.1146/annurev.fl.28.010196.002401 [32] DOI: 10.1103/RevModPhys.79.883 · Zbl 1205.76078 [33] DOI: 10.1002/cpa.3160050201 · Zbl 0046.41908 [34] New J. Phys. Fluids 9 pp 145– (2007) [35] Q. J. Mech. Appl. Maths 3 pp 140– (1950) [36] DOI: 10.1098/rsta.2010.0375 [37] DOI: 10.1017/S0022112091001659 [38] DOI: 10.1017/S0022112009008015 · Zbl 1183.76685 [39] DOI: 10.1017/S0022112006000656 · Zbl 1122.76003 [40] DOI: 10.1063/1.857540 [41] DOI: 10.1098/rspa.1952.0035 · Zbl 0046.18904 [42] DOI: 10.1088/0034-4885/72/9/096601 [43] DOI: 10.1103/PhysRevLett.77.4102 [44] Hydrodynamics (1932) · JFM 58.1298.05 [45] DOI: 10.1098/rsta.2010.0369 [46] DOI: 10.1063/1.3528260 · Zbl 06421588 [47] Mixed boundary value problems in potential theory (1966) · Zbl 0139.28801 [48] DOI: 10.1063/1.868619 · Zbl 1026.76551 [49] DOI: 10.1016/j.jcp.2004.10.014 · Zbl 1067.65088 [50] DOI: 10.1016/j.jcp.2006.11.007 · Zbl 1123.76044 [51] DOI: 10.1017/S002211200000313X · Zbl 1017.76024 [52] DOI: 10.1063/1.1850151 · Zbl 1187.76270 [53] DOI: 10.1146/annurev.fluid.35.101101.161213 · Zbl 1039.76028 [54] Boundary Layer Theory (1960) [55] Potential Flows of Viscous and Viscoelastic Fluids (2007) [56] DOI: 10.1146/annurev-fluid-121108-145558 [57] DOI: 10.1137/040604868 · Zbl 1108.76026 [58] Bull. Am. Phys. Soc (2005) [59] DOI: 10.1119/1.10903 [60] DOI: 10.1002/(SICI)1097-0363(19980915)28:3<501::AID-FLD730>3.0.CO;2-S · Zbl 0932.76065 [61] DOI: 10.1063/1.1476671 · Zbl 1185.76304 [62] DOI: 10.1002/fld.203 · Zbl 1007.76019 [63] Applied Hydro- and Aerodynamics (1957) [64] Intl J. Offshore Polar Engng 15 pp 1– (2005) [65] DOI: 10.1006/jcph.2000.6556 · Zbl 0977.76021 [66] J. Fluid Mech. 599 pp 111– (2008) [67] Low Reynolds Number Hydrodynamics (1965) [68] DOI: 10.1017/S0022112004000588 · Zbl 1131.76314 [69] Spectral Methods for Incompressible Viscous Flow (2001) [70] DOI: 10.1017/S0022112095000462 · Zbl 0847.76007
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