Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

P. D.  Towers; Z. Hussain; Paul Griffiths; Stephen Garrett

doi:10.1093/imamat/hxw041

Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

P. D. Towers, Z. Hussain, Paul Griffiths, Stephen Garrett

College of Engineering and Physical Sciences

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Abstract

We investigate the effects of compressibility and wall cooling on the stationary, viscous (Type II) instability mode within the 3D boundary layer over rotating cones with half-angle greater than 40∘ . The stationary mode is characterised by zero shear stress at the wall and a triple-deck solution is presented in the isothermal case. Asymptotic solutions are obtained which describe the structure of the wavenumber and the orientation of this mode as a function of local Mach number. It is found that a stationary mode is possible only over a finite range of local Mach number. Our conclusions are entirely consistent with the results of Seddougui 1990 , A nonlinear investigation of the stability models of instability of the trhee-dimensional Compresible boundary layer due to a rotating disc Q. J. Mech. Appl. Math. , 43, pt. 4. It is suggested that wall cooling has a significant stabilising effect, while reducing the half-angle is marginally destabilising. Solutions are presented for air.

Original language	English
Pages (from-to)	940–960
Number of pages	20
Journal	IMA Journal of Applied Mathematics
Volume	81
Issue number	5
Early online date	29 Aug 2016
DOIs	https://doi.org/10.1093/imamat/hxw041
Publication status	Published - 1 Oct 2016

Bibliographical note

© The author 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This is a pre-copyedited, author-produced version of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record is available online at: https://doi.org/10.1093/imamat/hxw041

Funding Information:
Engineering and Physical Sciences Research Council (grant EP/G061637/1 to S.J.G. and P.D.T.); Royal Academy of Engineering and the Leverhulme Trust (LSRF1415/11/29 to S.J.G and P.T.G.).

Keywords

Rotating cone
Type II
Compressible boundary-layer flow

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10.1093/imamat/hxw041

Towers_et_al_Viscous_modes_Journal_Applied_Mathematics
© The author 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This is a pre-copyedited, author-produced version of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record is available online at: https://doi.org/10.1093/imamat/hxw041
Accepted author manuscript, 988 KB

Cite this

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title = "Viscous modes within the compressible boundary-layer flow due to a broad rotating cone",

abstract = "We investigate the effects of compressibility and wall cooling on the stationary, viscous (Type II) instability mode within the 3D boundary layer over rotating cones with half-angle greater than 40∘ . The stationary mode is characterised by zero shear stress at the wall and a triple-deck solution is presented in the isothermal case. Asymptotic solutions are obtained which describe the structure of the wavenumber and the orientation of this mode as a function of local Mach number. It is found that a stationary mode is possible only over a finite range of local Mach number. Our conclusions are entirely consistent with the results of Seddougui 1990 , A nonlinear investigation of the stability models of instability of the trhee-dimensional Compresible boundary layer due to a rotating disc Q. J. Mech. Appl. Math. , 43, pt. 4. It is suggested that wall cooling has a significant stabilising effect, while reducing the half-angle is marginally destabilising. Solutions are presented for air.",

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author = "Towers, {P. D.} and Z. Hussain and Paul Griffiths and Stephen Garrett",

note = "{\textcopyright} The author 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This is a pre-copyedited, author-produced version of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record is available online at: https://doi.org/10.1093/imamat/hxw041 Funding Information: Engineering and Physical Sciences Research Council (grant EP/G061637/1 to S.J.G. and P.D.T.); Royal Academy of Engineering and the Leverhulme Trust (LSRF1415/11/29 to S.J.G and P.T.G.).",

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AU - Towers, P. D.

AU - Hussain, Z.

AU - Griffiths, Paul

AU - Garrett, Stephen

N1 - © The author 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This is a pre-copyedited, author-produced version of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record is available online at: https://doi.org/10.1093/imamat/hxw041 Funding Information: Engineering and Physical Sciences Research Council (grant EP/G061637/1 to S.J.G. and P.D.T.); Royal Academy of Engineering and the Leverhulme Trust (LSRF1415/11/29 to S.J.G and P.T.G.).

PY - 2016/10/1

Y1 - 2016/10/1

N2 - We investigate the effects of compressibility and wall cooling on the stationary, viscous (Type II) instability mode within the 3D boundary layer over rotating cones with half-angle greater than 40∘ . The stationary mode is characterised by zero shear stress at the wall and a triple-deck solution is presented in the isothermal case. Asymptotic solutions are obtained which describe the structure of the wavenumber and the orientation of this mode as a function of local Mach number. It is found that a stationary mode is possible only over a finite range of local Mach number. Our conclusions are entirely consistent with the results of Seddougui 1990 , A nonlinear investigation of the stability models of instability of the trhee-dimensional Compresible boundary layer due to a rotating disc Q. J. Mech. Appl. Math. , 43, pt. 4. It is suggested that wall cooling has a significant stabilising effect, while reducing the half-angle is marginally destabilising. Solutions are presented for air.

AB - We investigate the effects of compressibility and wall cooling on the stationary, viscous (Type II) instability mode within the 3D boundary layer over rotating cones with half-angle greater than 40∘ . The stationary mode is characterised by zero shear stress at the wall and a triple-deck solution is presented in the isothermal case. Asymptotic solutions are obtained which describe the structure of the wavenumber and the orientation of this mode as a function of local Mach number. It is found that a stationary mode is possible only over a finite range of local Mach number. Our conclusions are entirely consistent with the results of Seddougui 1990 , A nonlinear investigation of the stability models of instability of the trhee-dimensional Compresible boundary layer due to a rotating disc Q. J. Mech. Appl. Math. , 43, pt. 4. It is suggested that wall cooling has a significant stabilising effect, while reducing the half-angle is marginally destabilising. Solutions are presented for air.

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JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

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ER -

Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

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