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An adjoint approach for computing the receptivity of the rotating disc boundary layer to surface roughness

Published online by Cambridge University Press:  06 September 2021

Christian Thomas Open the ORCID record for Christian Thomas [Opens in a new window]  and
Christopher Davies Open the ORCID record for Christopher Davies [Opens in a new window]
Christian Thomas*
Affiliation:
Department of Mathematics and Statistics, Macquarie University, NSW2109, Australia
Christopher Davies
Affiliation:
School of Engineering, University of Leicester, University Road, LeicesterLE1 7RH, UK
*
†Email address for correspondence: [email protected]
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Abstract

An adjoint approach is developed to compute the receptivity of the rotating disc boundary layer to surface roughness. The adjoint linearised Navier–Stokes equations, in cylindrical coordinates, are derived and receptivity characteristics are computed for a broad range of azimuthal mode numbers using a fully equivalent velocity–vorticity formulation. For each set of flow conditions (i.e. azimuthal mode number), the adjoint method only requires that the linear and adjoint solutions be computed once. Thus, the adjoint approach offers significant computational and time advantages over alternative receptivity schemes (i.e. direct linearised Navier–Stokes) as they can be used to instantaneously compute the receptivity of boundary layer disturbances to many environmental mechanisms. Stationary cross-flow disturbances are established by randomly distributed surface roughness that is periodic in the azimuthal direction and modelled via a linearisation of the no-slip condition on the disc surface. Each roughness distribution is scaled on its respective root-mean-square. A Monte-Carlo type uncertainty quantification analysis is performed, whereby mean receptivity amplitudes are computed by averaging over many thousands of roughness realisations with variable length and wavelength filters. The amplitude of the cross-flow instability is significantly larger for roughness distributions near the conditions for neutral linear instability, while roughness elements radially outboard have a negligible effect on the receptivity process. Furthermore, receptivity increases sharply for roughness distributions that encompass wavelength scales equivalent to that associated with the cross-flow instability. Finally, mean receptivity characteristics are used to predict the radial range that stationary cross-flow vortices achieve amplitudes sufficient to invalidate the linear stability assumptions.

JFM classification

Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer stability Instability: Absolute/convective instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 926 , 10 November 2021 , A16
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Airiau, C. 2000 Non-parallel acoustic receptivity of a Blasius boundary layer using an adjoint approach. Flow Turbul. Combust. 65, 347–367.CrossRefGoogle Scholar
Appelquist, E., Schlatter, P., Alfredsson, P.H. & Lingwood, R.J. 2015 Global linear instability of the rotating-disk flow investigated through simulations. J. Fluid Mech. 765, 612–631.CrossRefGoogle Scholar
Appelquist, E., Schlatter, P., Alfredsson, P.H. & Lingwood, R.J. 2016 On the global nonlinear instability of the rotating-disk flow over a finite domain. J. Fluid Mech. 803, 332–355.CrossRefGoogle Scholar
Appelquist, E., Schlatter, P., Alfredsson, P.H. & Lingwood, R.J. 2018 Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices. J. Fluid Mech. 836, 43–71.CrossRefGoogle Scholar
Balachandar, S., Streett, C.L. & Malik, M.R. 1992 Secondary instability in rotating-disk flow. J. Fluid Mech. 242, 323–347.CrossRefGoogle Scholar
Briggs, R.J. 1964 Electron-Stream Interactions in Plasmas. MIT.CrossRefGoogle Scholar
Carpenter, M.H., Choudhari, M., Li, F., Streett, C.L. & Chang, C.L. 2010 Excitation of cross-flow instabilities in a swept wing boundary layer. AIAA Paper 2010-378.CrossRefGoogle Scholar
Chomaz, J.M., Huerre, P. & Redekopp, L.G. 1988 Bifurcations to local and global modes in spatially-developing flows. Phys. Rev. Lett. 60, 25–28.CrossRefGoogle ScholarPubMed
Choudhari, M. 1993 Boundary-layer receptivity due to distributed surface imperfections of a deterministic or random nature. Theor. Comput. Fluid Dyn. 4, 101–117.CrossRefGoogle Scholar
Choudhari, M. 1994 Roughness-induced generation of cross-flow vortices in three-dimensional boundary layers. Theor. Comput. Fluid Dyn. 6, 1–30.CrossRefGoogle Scholar
Choudhari, M. & Streett, C.L. 1992 A finite Reynolds-number approach for the prediction of boundary-layer receptivity in localized regions. Phys. Fluids A 4, 2495–2514.CrossRefGoogle Scholar
Collis, S.S. & Lele, S.K. 1999 Receptivity to surface roughness near a swept leading edge. J. Fluid Mech. 380, 141–168.CrossRefGoogle Scholar
Crouch, J.D. 1992 a Localized receptivity of boundary layers. Phys. Fluids A 4, 1408–1414.CrossRefGoogle Scholar
Crouch, J.D. 1992 b Non-localized receptivity of boundary layers. J. Fluid Mech. 244, 567–581.CrossRefGoogle Scholar
Davies, C & Carpenter, P.W. 2001 A novel velocity-vorticity formulation of the Navier–Stokes equations with applications to boundary layer disturbance evolution. J. Comput. Phys. 172, 119–165.CrossRefGoogle Scholar
Davies, C & Carpenter, P.W. 2003 Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. J. Fluid Mech. 486, 287–329.CrossRefGoogle Scholar
Dobrinsky, A. 2002 Adjoint analysis for receptivity prediction. PhD thesis, Rice University.Google Scholar
Dobrinsky, A. & Collis, S.S. 2000 Adjoint parabolized stability equations for receptivity prediction. AIAA Paper 2000-2651.CrossRefGoogle Scholar
Faller, A.J & Kaylor, R.E 1966 A numerical study of the instability of the laminar Ekman boundary-layer. J. Atmos. Sci. 23, 466–480.2.0.CO;2>CrossRefGoogle Scholar
Giannetti, F. & Luchini, P. 2006 Leading-edge receptivity by adjoint methods. J. Fluid Mech. 547, 21–53.CrossRefGoogle Scholar
Goldstein, M.E. 1983 The evolution of Tollmien–Schlichting waves near a leading edge. J. Fluid Mech. 127, 59–81.CrossRefGoogle Scholar
Goldstein, M.E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509–529.CrossRefGoogle Scholar
Gregory, N, Stuart, J.T. & Walker, W.S. 1955 On the stability of three-dimensional boundary-layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 248, 155–199.Google Scholar
Healey, J.J. 2010 Model for unstable global modes in the rotating-disk boundary layer. J. Fluid Mech. 663, 148–159.CrossRefGoogle Scholar
Hill, D.C. 1995 Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183–204.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Hunt, R.E & Crighton, D.G 1991 Instability of flows in spatially developing media. Proc. R. Soc. Lond. A 435, 109–128.Google Scholar
Imayama, S., Alfredsson, P.H. & Lingwood, R.J. 2012 A new way to describe the transition characteristics of a rotating-disk boundary-layer flow. Phys. Fluids 24, 031701.CrossRefGoogle Scholar
Imayama, S., Alfredsson, P.H. & Lingwood, R.J. 2013 An experimental study of edge effects on rotating-disk transition. J. Fluid Mech. 716, 638–657.CrossRefGoogle Scholar
Imayama, S., Alfredsson, P.H. & Lingwood, R.J. 2014 On the laminar-turbulent transition of the rotating-disk flow – the role of absolute instability. J. Fluid Mech. 745, 132–163.CrossRefGoogle Scholar
Imayama, S., Alfredsson, P.H. & Lingwood, R.J. 2016 Experimental study of rotating-disk boundary-layer flow with surface roughness. J. Fluid Mech. 786, 5–28.CrossRefGoogle Scholar
Jacobs, T.D.B., Junge, T. & Pastewka, L. 2017 Quantitative characterization of surface topography using spectral analysis. Surf. Topogr. Metrol. 5, 013001.CrossRefGoogle Scholar
von Kármán, Th. 1921 Über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1, 233–252.CrossRefGoogle Scholar
Kobayashi, R., Kohama, Y. & Takamadate, C. 1980 Spiral vortices in boundary layer transition regime on a rotating disk. Acta Mechanica 35, 71–82.CrossRefGoogle Scholar
Kohama, Y. 1984 Study on boundary layer transition of a rotating disk. Acta Mechanica 50, 193–199.CrossRefGoogle Scholar
Lee, K., Nishio, Y., Izawa, S. & Fukunishi, Y. 2017 Effects of location of excitation on the spiral vortices in the transitional region of a rotating-disk flow. Phys. Fluids 29, 084106.CrossRefGoogle Scholar
Lee, K., Nishio, Y., Izawa, S. & Fukunishi, Y. 2018 The effect of downstream turbulent region on the spiral vortex structures of a rotating-disk flow. J. Fluid Mech. 844, 274–296.CrossRefGoogle Scholar
Lingwood, R.J. & Alfredsson, P.H. 2015 Instabilities of the von Kármán boundary layer. Appl. Mech. Rev. 67, 030803.CrossRefGoogle Scholar
Lingwood, R.J. 1995 Absolute instability of the boundary-layer on a rotating-disk. J. Fluid Mech. 299, 17–33.CrossRefGoogle Scholar
Lingwood, R.J. 1996 An experimental study of absolute instability of the rotating-disk boundary-layer flow. J. Fluid Mech. 314, 373–405.CrossRefGoogle Scholar
Lingwood, R.J. 1997 On the effects of suction and injection on the absolute instability of the rotating-disk boundary layers. Phys. Fluids 9, 1317–1328.CrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 1998 Görtler vortices: a backward-in-time approach to the receptivity problem. J. Fluid Mech. 363, 1–23.CrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493517.CrossRefGoogle Scholar
Mack, L.M. 1985 The wave pattern produced by point source on a rotating disk. AIAA Paper 85-0490.CrossRefGoogle Scholar
Malik, M.R. 1986 The neutral curve for stationary disturbances in rotating-disk flow. J. Fluid Mech. 164, 275–287.CrossRefGoogle Scholar
Malik, M.R., Wilkinson, S. & Orszag, S.A. 1981 Instability and transition in rotating disk flow. AIAA J. 19, 1131–1138.CrossRefGoogle Scholar
Morkovin, M.V. 1969 On the many faces of transition. In Viscous Drag Reduction: Proceedings of the Symposium on Viscous Drag Reduction held at the LTV Research Center, Dallas, Texas, September 24 and 25, 1968 (ed. C.S. Wells), pp. 1–31. Springer.CrossRefGoogle Scholar
Mughal, S.M. & Ashworth, R. 2013 Uncertainty quantification based receptivity modelling of cross-flow instabilities induced by distributed surface roughness in swept wing boundary layers. 43rd AIAA Fluid Dynamics Conference, 2013. AIAA Paper 2013-3106.Google Scholar
Ng, L.L. & Crouch, J.D. 1999 Roughness-induced receptivity to cross-flow vortices on a swept wing. Phys. Fluids 11, 432.CrossRefGoogle Scholar
Othman, H. & Corke, T.C. 2006 Experimental investigation of absolute instability of a rotating-disk boundary-layer. J. Fluid Mech. 565, 63–94.CrossRefGoogle Scholar
Pier, B. 2003 Finite-amplitude cross-flow vortices, secondary instability and transition in the rotating-disk boundary layer. J. Fluid Mech. 487, 315–343.CrossRefGoogle Scholar
Pier, B. 2013 Transition near the edge of a rotating disk. J. Fluid Mech. 737, R1.CrossRefGoogle Scholar
Pralits, J.O. & Hanifi, A. 2003 Optimization of steady suction for disturbance control on infinite swept wings. Phys. Fluids 15, 2756–2772.CrossRefGoogle Scholar
Pralits, J.O., Hanifi, A. & Henningson, D.S. 2002 Adjoint-based optimisation of steady suction for disturbance control in incompressible flows. J. Fluid Mech. 467, 129–161.CrossRefGoogle Scholar
Radeztsky, R.H., Reibert, M.S. & Saric, W.S. 1999 Effect of isolated micronsized roughness on transition in swept-wing flows. AIAA J. 37, 1370–1377.CrossRefGoogle Scholar
Raposo, H., Mughal, S. & Ashworth, R. 2018 Acoustic receptivity and transition modeling of Tollmien–Schlichting disturbances induced by distributed surface roughness. Phys. Fluids 30, 044105.CrossRefGoogle Scholar
Reibert, M.S., Saric, W.S., Carillo, R.B. & Chapman, K.L. 1996 Experiments in nonlinear saturation of stationary cross-flow vortices in a swept-wing boundary layer. AIAA Paper 96-0184.Google Scholar
Ruban, A.I. 1984 On Tollmien–Schlichting wave generation by sound. In Laminar-Turbulent Transition. International Union of Theoretical and Applied Mechanics (ed. V.V. Kozlov), pp. 313–320, Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Saric, W.S., Reed, H.L. & Kerschen, E.J. 2002 Boundary layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34, 291–319.CrossRefGoogle Scholar
Saric, W.S., Reed, H.L. & White, E.B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440.CrossRefGoogle Scholar
Saric, W.S., Carrillo, R.B. Jr. & Reibert, M.S. 1998 Leading-edge roughness as a transition control mechanism. AIAA Paper 98-0781.CrossRefGoogle Scholar
Sayles, S. & Thomas, T.R. 1978 Surface topography as a nonstationary random process. Nature 271, 431–434.CrossRefGoogle Scholar
Schrader, L.U., Amin, S. & Brandt, L. 2010 Transition to turbulence in the boundary layer over a smooth and rough swept plate exposed to free-stream turbulence. J. Fluid Mech. 646, 297–325.CrossRefGoogle Scholar
Schrader, L.U., Brandt, L. & Henningson, D.S. 2009 Receptivity mechanisms in three-dimensional boundary layer flows. J. Fluid Mech. 618, 209–241.CrossRefGoogle Scholar
Siddiqui, M.E., Mukund, V., Scott, J. & Pier, B. 2013 Experimental characterization of transition region in rotating-disk boundary layer. Phys. Fluids 25, 034102.CrossRefGoogle Scholar
Tempelmann, D., Hanifi, A. & Henningson, D.S. 2012 a Swept-wing boundary layer receptivity. J. Fluid Mech. 700, 490–501.CrossRefGoogle Scholar
Tempelmann, D., Schrader, L.U., Hanifi, A., Brandt, L. & Henningson, D.S. 2012 b Swept wing boundary-layer receptivity to localized surface roughness. J. Fluid Mech. 711, 516–544.CrossRefGoogle Scholar
Thomas, C. & Davies, C. 2018 On the impulse response and global instability development of the infinite rotating-disc boundary layer. J. Fluid Mech. 857, 239–269.CrossRefGoogle Scholar
Thomas, C. & Davies, C. 2019 Global linear instability of rotating-cone boundary layers in a quiescent medium. Phys. Rev. Fluids 4, 043902.CrossRefGoogle Scholar
Thomas, C., Mughal, S. & Ashworth, R. 2017 On predicting receptivity to surface roughness in a compressible infinite swept wing boundary layer. Phys. Fluids 29, 034102.CrossRefGoogle Scholar
Thomas, C., Stephen, S.O. & Davies, C. 2020 Effects of partial slip on the local-global linear stability of the rotating disk boundary layer. Phys. Fluids 32, 074105.CrossRefGoogle Scholar
Van Deusen, D. 1967 A statistical technique for the dynamic analysis of vehicles traversing rough yielding and non-yielding surfaces. NASA Tech. Rep. CR-659, 178 pages.Google Scholar
Wilkinson, S. & Malik, M.R. 1985 Stability experiments in the flow over a rotating disk. AIAA J. 23, 588–595.CrossRefGoogle Scholar
Wu, X. 2001 a On local boundary-layer receptivity to vortical disturbances in the free stream. J. Fluid Mech. 449, 373–393.CrossRefGoogle Scholar
Wu, X. 2001 b Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances: a second-order asymptotic theory and comparison with experiments. J. Fluid Mech. 431, 91–133.CrossRefGoogle Scholar