There are two primary aims of this paper: the first aim is to investigate the effects of the roughness types of the Miklavčič and Wang model on stationary disturbances of the boundary-layer flow over a broad rotating cone in still fluid. The second aim is to examine similar effects of surface roughness, but on non-stationary modes of crossflow instability. This study begins with the formulations of the mean-flow system based on the cone geometry. These equations are solved using a spectral numerical method based on Chebyshev polynomials and then used to formulate the linear stability system, which are computed for obtaining neutral curves of the unsteady flows. For the stationary modes, our results indicate that the inviscid instability (type I mode) is more stable, while the viscous instability (type II mode) entirely eliminates, as concentric grooves or isotropic roughness and the cone half-angle increase. In contrast, streamwise grooves have a slight stabilizing effect on the type I mode and a significant destabilizing effect on the viscous instability. Another finding indicates that decreasing the half-angle leads to a greater stabilizing effect of isotropic roughness on the type I modes. Our outcomes are also confirmed by the growth rate and the energy analysis, which shows a large reduction of the total energy balance as a result of increasing concentric grooves or isotropic roughness for the crossflow mode. For non-stationary modes, similar effects are observed in that increasing all levels of roughness stabilizes the type I branch (with concentric grooves and isotropic roughness having a much stronger effect than streamwise grooves), and although increasing concentric grooves and isotropic roughness stabilizes the type II branch, increasing streamwise grooves destabilizes this branch. For modes traveling slower than the rotation of the cone, concentric grooves generally have a stronger stabilization effect for the type I branch, whereas for modes traveling faster than the cone, isotropic grooves have a stronger stabilization effect on the type I branch for all broad half angles. Importantly, increasing concentric grooves and isotropic roughness increases the frequency at which the most dangerous modes occur, whereas increasing the streamwise grooves reduces the frequency at which the most dangerous modes occur.

1.
T.
von Kármán
,
Über Laminare Und Turbulente Reibung
(
Z. Angew. Math. Mech
,
1921
).
2.
H. W.
Liepmann
and
G. L.
Brown
, “
Active control of laminar-turbulent transition
,”
J. Fluid Mech.
118
,
201
(
1982
).
3.
M.
Miklavčič
and
C.
Wang
, “
The flow due to a rough rotating disk
,”
Z. Angew. Math. Phys.
55
,
235
246
(
2004
).
4.
T.
Wiegand
and
H.
Bestek
, “
Transition process of a wave train in a laminar boundary layer
,” in
New Results in Numerical and Experimental Fluid Mechanics
(
Springer
,
1997
), Vol.
60
.
5.
M. A. S.
Al-Malki
,
Z.
Hussain
,
S. J.
Garrett
, and
S. W.
Calabretto
, “
Effects of parietal suction and injection on the stability of the blasius boundary-layer flow over a permeable, heated plate
,”
Phys. Rev. Fluid
6
,
113902
(
2021
).
6.
F.
Kreith
,
D.
Ellis
, and
J.
Giesing
, “
An experimental investigation of the flow engendered by a rotating cone
,”
Appl. Sci. Res.
11
,
430
440
(
1963
).
7.
C. L.
Tein
and
D. T.
Campbell
, “
Heat and mass transfer from rotating cones
,”
J. Fluid Mech.
17
,
105
112
(
1963
).
8.
R.
Kappesser
,
R.
Greif
, and
I.
Cornet
, “
Mass transfer on rotating cones
,”
Appl. Sci. Res.
28
,
442
452
(
1973
).
9.
F.
Salzberg
and
S. P.
Kezios
, “
Mass transfer from a rotating cone in axisymmetric flow
,”
J. Heat Transfer
87
,
469
476
(
1965
).
10.
L.
Mack
, “
The wave pattern produced by a point source on a rotating disk
,” AIAA Paper No. 1985-490,
1985
.
11.
R.
Lingwood
, “
An experimental study of absolute instability of the rotating-disk boundary-layer flow
,”
J. Fluid Mech.
314
,
373
405
(
1996
).
12.
B.
Fedorov
,
G.
Plavnik
,
I.
Prokhorov
, and
L.
Zhukhovitskii
, “
Transitional flow conditions on a rotating disk
,”
J. Eng. Phys. Thermophys.
31
,
1448
1453
(
1976
).
13.
R.
Kobayashi
,
Y.
Kohama
, and
M.
Kurosawa
, “
Boundary-layer transition on a rotating cone in axial flow
,”
J. Fluid Mech.
127
,
341
352
(
1983
).
14.
R.
Kobayashi
and
H.
Izumi
, “
Boundary-layer transition on a rotating cone in still fluid
,”
J. Fluid Mech.
127
,
353
364
(
1983
).
15.
Y.
Kohama
, “
Flow structures formed by axisymmetric spinning bodies
,”
AIAA J.
23
,
1445
1447
(
1985
).
16.
T. J.
Mueller
,
R. C.
Nelson
,
J. T.
Kegelman
, and
M. V.
Morkovin
, “
Smoke visualisation of boundary-layer transition on a spinning axisymmetric body
,”
AIAA J.
19
,
1607
1608
(
1981
).
17.
S. J.
Garrett
, “
The stability and transition of the boundary layer on rotating bodies
,” Ph.D. thesis (
Cambridge University
,
2002
).
18.
S. J.
Garrett
and
N.
Peake
, “
The absolute instability of the boundary layer on a rotating cone
,”
Eur. J. Mech. B
26
,
344
353
(
2007
).
19.
Z.
Hussain
, “
Stability and transition of three-dimensional rotating boundary layers
,” Ph.D. thesis (
University of Birmingham
,
2010
).
20.
A. J.
Colley
,
P. J.
Thomas
,
P. W.
Carpenter
, and
A. J.
Cooper
, “
An experimental study of boundary-layer transition over a rotating, compliant disk
,”
Phys. Fluids
11
,
3340
3352
(
1999
).
21.
A. J.
Colley
,
P.
Carpenter
,
P. J.
Thomas
,
R.
Ali
, and
F.
Zoueshtiagh
, “
Experimental verification of the type-II-eigenmode destabilization in the boundary layer over a compliant rotating disk
,”
Phys. Fluids
18
,
054107
(
2006
).
22.
T.
Watanabe
,
H. M.
Warui
, and
N.
Fujisawa
, “
Effect of distributed roughness on laminar-turbulent transition in the boundary layer over a rotating cone
,”
Exp. Fluids
14
,
390
392
(
1993
).
23.
A. J.
Cooper
,
J. H.
Harris
,
S. J.
Garrett
,
M.
Özkan
, and
P. J.
Thomas
, “
The effect of anisotropic and isotropic roughness on the convective stability of the rotating disk boundary layer
,”
Phys. Fluids
27
,
014107
(
2015
).
24.
P.
Balakumar
and
M.
Malik
, “
Traveling disturbances in rotating-disk flow
,”
Theor. Comput. Fluid Dyn.
2
,
125
137
(
1990
).
25.
M.
Turkyilmazoglu
, “
Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk
,”
Q. Appl. Math.
65
,
43
68
(
2007
).
26.
Z.
Hussain
,
S. J.
Garrett
, and
S. O.
Stephen
, “
The instability of the boundary layer over a disk rotating in an enforced axial flow
,”
Phys. Fluids
23
,
114108
(
2011
).
27.
S. J.
Garrett
, “
Linear growth rates of types i and ii convective modes within the rotating-cone boundary layer
,”
Fluid Dyn. Res.
42
,
025504
(
2010
).
28.
M.
Fildes
,
Z.
Hussain
,
J.
Unadkat
, and
S. J.
Garrett
, “
Analysis of boundary layer flow over a broad rotating cone in still fluid with non-stationary modes
,”
Phys. Fluids
32
,
124118
(
2020
).
29.
M. S.
Yoon
,
J. M.
Hyun
, and
J. S.
Park
, “
Flow and heat transfer over a rotating disk with surface roughness
,”
Int. J. Heat Fluid Flow
28
,
262
267
(
2007
).
30.
S. J.
Garrett
,
A. J.
Cooper
,
J. H.
Harris
,
M.
Özkan
,
A.
Segalini
, and
P. J.
Thomas
, “
On the stability of von Kármán rotating-disk boundary layers with radial anisotropic surface roughness
,”
Phys. Fluids
28
,
014104
014158
(
2016
).
31.
S. J.
Garrett
,
J.
Harris
, and
P. J.
Thomas
, “
On the effect of distributed roughness on transition over rotor-stator devices
,” in
Proceedings of the International Council of the Aeronautical Sciences 2012
Brisbane, Australia (ICAS, 2012), p. 18.
32.
B.
Alveroğlu
,
A.
Segalini
, and
S. J.
Garrett
, “
The effect of surface roughness on the convective instability of the BEK family of boundary-layer flows
,”
Eur. J. Mech. B
56
,
178
187
(
2016
).
33.
A. A.
Alqarni
,
B.
Alveroğlu
,
P. T.
Griffiths
, and
S. J.
Garrett
, “
The instability of non-Newtonian boundary-layer flows over rough rotating disks
,”
J. Non-Newtonian Fluid Mech.
273
,
104174
(
2019
).
34.
M. A. S.
Al-Malki
,
S. J.
Garrett
,
S.
Camarri
, and
Z.
Hussain
, “
The effects of roughness levels on the instability of the boundary-layer flow over a rotating disk with an enforced axial flow
,”
Phys. Fluids
33
,
104109
(
2021
).
35.
S.
Garrett
,
Z.
Hussain
, and
S. O.
Stephen
, “
The cross-flow instability of the boundary layer on a rotating cone
,”
J. Fluid Mech.
622
,
209
232
(
2009
).
36.
Z.
Hussain
,
S. J.
Garrett
, and
S. O.
Stephen
, “
The centrifugal instability of the boundary-layer flow over slender rotating cones
,”
J. Fluid Mech.
755
,
274
293
(
2014
).
37.
S.
Tambe
,
F.
Schrijer
,
A. G.
Rao
, and
L.
Veldhuis
, “
An experimental method to investigate coherent spiral vortices in the boundary layer over rotating bodies of revolution
,”
Exp. Fluids
60
,
115
(
2019
).
38.
C. S.
Wu
, “
The three dimensional incompressible laminar boundary layer on a rotating cone
,”
Appl. Sci. Res.
8
,
140
146
(
1959
).
39.
C. L.
Tein
, “
Heat transfer by laminar flow from a rotating cone
,”
ASME J. Heat Transfer
82
,
252
253
(
1960
).
40.
F. T.
Smith
, “
Laminar flow over a small hump on a flat plate
,”
J. Fluid Mech.
57
,
803
824
(
1973
).
41.
C.
Chicchiero
,
A.
Segalini
, and
S.
Camarri
, “
Triple-deck analysis of the steady flow over a rotating disk with surface roughness
,”
Phys. Rev. Fluid
6
,
014103
(
2021
).
42.
Z.
Hussain
and
S. J.
Garrett
, “
On the stability of boundary-layer flow over a rotating cone using new solution methods
,”
J. Phys.: Conf. Ser.
1909
,
012041
(
2021
).
43.
A. J.
Cooper
and
P. W.
Carpenter
, “
The stability of rotating-disc boundary layer flow over a compliant wall. I. Type I and II instabilities
,”
J. Fluid Mech.
350
,
231
259
(
1997
).
You do not currently have access to this content.