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The turbulence cascade in the near wake of a square prism

  • F. Alves Portela (a1), G. Papadakis (a1) and J. C. Vassilicos (a1)

We present a study of the turbulence cascade on the centreline of an inhomogeneous and anisotropic near-field turbulent wake generated by a square prism at a Reynolds number of $Re=3900$ using the Kármán–Howarth–Monin–Hill equation. This is the fully generalised scale-by-scale energy balance which, unlike the Kármán–Howarth equation, does not require homogeneity or isotropy assumptions. Our data are obtained from a direct numerical simulation and therefore enable us to access all of the processes involved in this energy balance. A significant range of length scales exists where the orientation-averaged nonlinear interscale transfer rate is approximately constant and negative, indicating a forward turbulence cascade on average. This average cascade consists of coexisting forward and inverse cascade behaviours in different scale-space orientations. With increasing distance from the prism but within the near field of the wake, the orientation-averaged nonlinear interscale transfer rate tends to be approximately equal to minus the turbulence dissipation rate even though all of the inhomogeneity-related energy processes in the scale-by-scale energy balance are significant, if not equally important. We also find well-defined near $-5/3$ energy spectra in the streamwise direction, in particular at a centreline position where the inverse cascade behaviour occurs for streamwise oriented length scales.

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R. A. Antonia , R. J. Smalley , T. Zhou , F. Anselmet  & L. Danaila 2003 Similarity of energy structure functions in decaying homogeneous isotropic turbulence. J. Fluid Mech. 487, 245269.

P. W. Bearman  & E. D. Obasaju 1982 An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. J. Fluid Mech. 119, 297321.

L. H. Benedict  & R. D. Gould 1996 Towards better uncertainty estimates for turbulence statistics. Exp. Fluids 22 (2), 129136.

M. S. Bloor  & J. H. Gerrard 1966 Measurements on turbulent vortices in a cylinder wake. Proc. R. Soc. Lond. A 294 (1438), 319342.

M. Braza , R. Perrin  & Y. Hoarau 2006 Turbulence properties in the cylinder wake at high Reynolds numbers. J. Fluids Struct. 22 (6–7), 757771.

B. J. Cantwell  & D. Coles 1983 An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech. 136, 321.

I. P. Castro 2016 Dissipative distinctions. J. Fluid Mech. 788, 14.

F. H. Champagne , V. G. Harris  & S. Corrsin 1970 Experiments on nearly homogeneous turbulent shear flow. J. Fluid Mech. 41 (01), 81.

J. M. Chen  & C. H. Liu 1999 Vortex shedding and surface pressures on a square cylinder at incidence to a uniform air stream. Intl J. Heat Fluid Flow 20 (6), 592597.

D. F. G. Durão , M. V. Heitor  & J. C. F. Pereira 1988 Measurements of turbulent and periodic flows around a square cross-section cylinder. Exp. Fluids 6 (5), 298304.

U. Frisch 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.

W. K. George 1978 Processing of random signals. In Dynamic Flow Conference on Dynamic Measurements in Unsteady Flows (ed. B. W. Hansen ), pp. 757800. Springer.

R. Gomes-Fernandes , B. Ganapathisubramani  & J. C. Vassilicos 2015 The energy cascade in near-field non-homogeneous non-isotropic turbulence. J. Fluid Mech. 771, 676705.

S. Goto  & J. C. Vassilicos 2015 Energy dissipation and flux laws for unsteady turbulence. Phys. Lett. A 379 (16–17), 11441148.

S. Goto  & J. C. Vassilicos 2016 Local equilibrium hypothesis and Taylor’s dissipation law. Fluid Dyn. Res. 48 (2), 021402.

R. J. Hearst  & P. Lavoie 2014 Scale-by-scale energy budget in fractal element grid-generated turbulence. J. Turbul. 15 (8), 540554.

R. J. Hill 2001 Equations relating structure functions of all orders. J. Fluid Mech. 434, 379388.

R. J. Hill 2002a Exact second-order structure-function relationships. J. Fluid Mech. 468, 317326.

J. C. Hu , Y. Zhou  & C. Dalton 2006 Effects of the corner radius on the near wake of a square prism. Exp. Fluids 40 (1), 106118.

R. I. Issa 1986 Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62 (1), 4065.

R. H. Kraichnan 1974 On Kolmogorov’s inertial-range theories. J. Fluid Mech. 62 (02), 305.

A. G. Kravchenko  & P. Moin 2000 Numerical studies of flow over a circular cylinder at Re D = 3900. Phys. Fluids 12 (2), 403.

S. Laizet , J. Nedić  & J. C. Vassilicos 2015 The spatial origin of - 5/3 spectra in grid-generated turbulence. Phys. Fluids 27 (6), 065115.

S. Laizet , J. C. Vassilicos  & C. Cambon 2013 Interscale energy transfer in decaying turbulence and vorticity–strain-rate dynamics in grid-generated turbulence. Fluid Dyn. Res. 45 (6), 061408.

O. Lehmkuhl , I. Rodríguez , R. Borrell  & A. Oliva 2013 Low-frequency unsteadiness in the vortex formation region of a circular cylinder. Phys. Fluids 25 (8), 085109.

M. Lesieur 2008 Turbulence in Fluids, 4th edn. Springer.

E. Lindborg 1996 A note on Kolmogorov’s third-order structure-function law, the local isotropy hypothesis and the pressure–velocity correlation. J. Fluid Mech. 326, 343356.

E. Lindborg 1999 Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech. 388 (1999), 259288; S0022112099004851.

J. L. Lumley 1965 Interpretation of time spectra measured in high-intensity shear flows. Phys. Fluids 8 (6), 1056.

D. A. Lyn , S. Einav , W. Rodi  & J. H. Park 1995 A laser-Doppler velocimetry study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder. J. Fluid Mech. 304, 285.

X. Ma , G. S. Karamanos  & G. E. Karniadakis 2000 Dynamics and low-dimensionality of a turbulent near wake. J. Fluid Mech. 410, 2965.

W. D. McComb 2014 Homogeneous, Isotropic Turbulence. Oxford University Press.

G. Melina , P. J. K. Bruce  & J. C. Vassilicos 2016 Vortex shedding effects in grid-generated turbulence. Phys. Rev. Fluids 1 (4), 044402.

J. Nedić , S. Tavoularis  & I. Marusic 2017 Dissipation scaling in constant-pressure turbulent boundary layers. Phys. Rev. Fluids 2 (3), 032601.

C. Norberg 1993 Flow around rectangular cylinders: pressure forces and wake frequencies. J. Wind Engng Ind. Aerodyn. 49 (1–3), 187196.

M. Obligado , T. Dairay  & J. C. Vassilicos 2016 Nonequilibrium scalings of turbulent wakes. Phys. Rev. Fluids 1 (4), 044409.

L. Ong  & J. Wallace 1996 The velocity field of the turbulent very near wake of a circular cylinder. Exp. Fluids 20 (6), 441453.

S. B. Pope 2000 Turbulent Flows. Cambridge University Press.

M. M. Rogers  & R. D. Moser 1994 Direct simulation of a self-similar turbulent mixing layer. Phys. Fluids 6 (2), 903.

A. Sohankar 2006 Flow over a bluff body from moderate to high Reynolds numbers using large eddy simulation. Comput. Fluids 35 (10), 11541168.

A. Sohankar , C. Norberg  & L. Davidson 1999 Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers. Phys. Fluids 11 (2), 288.

F. Thiesset , R. A. Antonia  & L. Danaila 2013a Restricted scaling range models for turbulent velocity and scalar energy transfers in decaying turbulence. J. Turbul. 14 (3), 2541.

F. Thiesset , L. Danaila  & R. A. Antonia 2013b Dynamical effect of the total strain induced by the coherent motion on local isotropy in a wake. J. Fluid Mech. 720, 393423.

F. Thiesset , L. Danaila  & R. A. Antonia 2016 Dynamical interactions between the coherent motion and small scales in a cylinder wake. J. Fluid Mech. 749, 201226.

F. X. Trias , A. Gorobets  & A. Oliva 2015 Turbulent flow around a square cylinder at Reynolds number 22 000: a DNS study. Comput. Fluids 123 (22), 8798.

A. Tsinober 2009 An Informal Conceptual Introduction to Turbulence, 2nd edn. Springer.

M. S. Uberoi  & P. Freymuth 1969 Spectra of turbulence in wakes behind circular cylinders. Phys. Fluids 12 (7), 1359.

P. C. Valente  & J. C. Vassilicos 2015 The energy cascade in grid-generated non-equilibrium decaying turbulence. Phys. Fluids 27 (4), 045103.

J. C. Vassilicos 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47 (1), 95114.

P. Voke 1996 Flow past a square cylinder: test case LES2. In Direct and Large Eddy Simulation II, vol. 5, pp. 355373. Springer.

J. G. Wissink  & W. Rodi 2008 Numerical study of the near wake of a circular cylinder. Intl J. Heat Fluid Flow 29 (4), 10601070.

J. C. Wyngaard  & S. F. Clifford 1977 Taylor’s hypothesis and high-frequency turbulence spectra. J. Atmos. Sci. 34 (6), 922929.

Y. Zhou  & R. A. Antonia 1992 Convection velocity measurements in a cylinder wake. Exp. Fluids 13, 6370.

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