Proceedings of the 8th International and 47th National Conference on Fluid Mechanics and Fluid Power (FMFP)
December 09-11, 2020, IIT Guwahati, Guwahati-781039, Assam, India
FMFP2020–094
The Effect of the Forebody Shapes on the Stability of the Axisymmetric Boundary Layer
Parth Pandya1, Ramesh Bhoraniya2, Ravi Kant3
1, 2
3
Department of Mechanical Engineering, Marwadi Education Foundations Group of Institutions, Rajkot-360003, India
Department of Mechanical Engineering, SOT, Pandit Deendayal Petroleum University, Gandhinagar-382007, India
ABSTRACT
two steps. In the first step, base flow computation was done,
and in the second step, the stability analysis was performed.
The base flow computations were done using Finite Volume
code ANSYS Fluent. The stability analysis was performed by
solving a general eigenvalues problem. Three different
axisymmetric forebody shapes named ellipsoid, paraboloid, and
sharp-cone with different fineness ratio (FR) of 2.5, 5, and 7.5
were considered. The effects of these forebody shapes have
been studied on the axisymmetric and helical modes.
This paper represents the effect of the axisymmetric forebody
shapes on the stability of an axisymmetric boundary layer
developed on a circular cylinder for incompressible flow. We
considered sharp-cone, paraboloid, and ellipsoid with fineness
ratio (FR) of 2.5, 5, and 7.5. The fineness ratio is defined by the
ratio of the length of forebody shapes to the diameter of the
cylinder. The incoming base flow velocity is parallel to the
cylinder's axis, and hence it is axisymmetric. The parallel base
flow is considered, and spatial local stability analysis is
performed. The streamwise wave-number () is complex, and
wave frequency () is real. The Spectral collocation method
was used to discretize the stability equations. The discretized
equations, together with the boundary conditions, form a
polynomial eigenvalue problem. To understand the effect of
forebody shapes and FR, the spatial growth rate (-i) is
computed at different streamwise locations at given critical
Reynolds number (Re) 12439 and 6070 for axisymmetric and
non-axisymmetric modes respectively. It is found that small
disturbances are more stable for the sharp-cone and least stable
for the ellipsoid. The three different Reynolds number 1000,
4000 and 10000 are considered to understand that for which Re
the flow is stable for a given particular shape.
2. LITERATURE REVIEW AND OBJECTIVE
Tutty and Price [1] have worked on Boundary layer flow on a
long thin cylinder. They studied the boundary layer
characteristics and stability in case of the flow over a long
cylinder. They found that for the axisymmetric mode (n = 0),
the critical Reynolds number is 12439, and for the helical mode
(n = 1), it is 1060. Vinod et al. (2002) [2] have reviewed on
Stability Analysis of an Axisymmetric Boundary Layer. This
research paper shows that transverse curvature has a significant
effect on the stability of the boundary layer. They found that for
high curvature, the helical mode is dominant and unstable.
Vinod and Govindarajan (2006) [3] have worked on Linear and
secondary instabilities in incompressible axisymmetric
boundary layers with the effect of the transverse curvature.
They found that axisymmetric mode becomes unstable above
the critical Reynolds number of 12439, and the helical mode is
never unstable for curvature above unity. Glauert and Light-hill
(1954) [4] have worked on the Axisymmetric boundary layer
on a long thin cylinder. This paper represents the investigation
of the laminar boundary layer in axial flow about a long thin
cylinder. Mérida, Jelliti, and Lili (2016) [5] worked on Laminar
instability of parallel and nonparallel flows for the adiabatic flat
plate. It represents the effect of Mach No. on the stability of the
flow. They found that the critical Reynolds number of
nonparallel flow decreasing with increasing Mach No. while the
critical Reynolds number of parallel flow increasing with
increasing Mach No. Wan, Yang and Zhou and sun (2014) [6]
have worked on Linear stability analysis of supersonic
axisymmetric jets. This paper represents the study of linear
stability of axisymmetric jets with different velocity,
momentum thickness, and core temperature. They found that
increased velocity and core temperature would increase the
disturbance amplification, and hence the flow becomes
unstable. Muralidhar and Govindarajan (2015) [7] have worked
Keywords: Forebody, Axisymmetric boundary layer, Local
stability, spatial stability, Reynolds number
1. INTRODUCTION
Hydrodynamic stability is an important theory in fluid
dynamics. It includes the stability analysis of the fluid flow and
its theoretical approaches. Hydrodynamic stability has
significant
importance
in
aerodynamics,
marine
hydrodynamics, fluid mixing process, and flow-control
methods. This paper presents the linear spatial stability analysis
of an axisymmetric boundary layer under the effect of forebody
shapes on the boundary layer's stability. When fluid flows over
the flat surface (i.e., flat plate), it generates the plane boundary
layer over the flat surface.
Similarly, when fluid flows over the axisymmetric body like a
cylinder, it generates the axisymmetric boundary layer over the
surface. Most of the underwater vehicles consist of a primary
cylindrical surface, which forms the axisymmetric boundary
layer when fluid flows over it. The underwater vehicles also
have specific forebody shapes that impact the stability of an
axisymmetric boundary layer. The present study follows the
1
on Linear stability analysis of the boundary layer over a long
cylinder with rotation. This paper represents the stability
analysis of a cylinder, which rotates about its axis with axial
flow. They found that as the rotation increases, the critical
Reynolds number will decrease; hence flow will become
unstable. Kumar and Mahesh (2018) [8] have worked on the
analysis of the axisymmetric boundary layers. In this article,
axisymmetric boundary layers were studied using an integral
analysis of the governing equations for axial flow over a
circular cylinder. They found that the presence of transverse
curvature increases the skin-friction coefficient (Cf). LIU,
WANG, and PIAO (2016) [9] have worked on Linear stability
analysis of interactions between mixing layer and boundary
layer flows. This paper represents the linear stability analysis of
incompressible boundary layer flows. Three Mixing layers are
mentioned in this paper: - (1) (Wake + Shear) layer + boundary
layer (WSBL) (2) Wake layer + Boundary layer (WBL) (3)
Shear layer + Boundary layer (SBL). They found that critical
Reynolds numbers for the WBL, SBL, & WSBL are 481,492
&, 497 respectively. Xin, Dengbin (2009) [10] have worked on
the Nonlinear Stability of Supersonic Nonparallel Boundary
Layer Flows. This paper represents a nonlinear study of
disturbance waves in supersonic nonparallel boundary layer
flows. Johnson, Pinarbasi (2014) [11] have worked on ‘The
effect of pressure gradient on boundary layer receptivity. They
found that APG leads towards receptivity, which is more than
double for the FPG.
The present study's main objective is to understand the effect of
different forebody shapes like sharp-cone, ellipsoid, and
paraboloid on the stability of the axisymmetric boundary layer.
The spatial growth rate has been computed for different shapes
and FR at different streamwise locations for a given critical
Reynolds numbers.
In Figure 1, the curve p-a is a circular arc which is an inlet of
flow domain and inlet boundary condition is applied to it.
The Reynolds number was calculated considering the body
radius of the cylinder (a) as a characteristic length.
Re =
U  a
(1)

Where , U, and  are density, free-stream velocity, and
dynamic viscosity of the fluid, respectively. The stability
analysis was performed considering the parallel flow
assumption of modified base flow due to the forebody (bof).
The governing stability equations have been derived in the
cylindrical polar coordinates (x, r, ). The normal mode form
of the disturbances was considered. The perturbations
equations can be written as,
q( x, r , , t ) = qˆ (r )ei( x+n −t ) ,
Where, q = [u, v, w, p].
The stability equations were derived using standard procedure
and are as follow,
u
t
v
t
w
t
u
x
The enhanced understanding and knowledge will be useful in
deciding appropriate flow control strategy and thus to design
and operate energy efficient vehicles with reduced emission.
+v
+U
U
r
v
x
+U
+
v
r
+U
=−
w
x
+
u
x
p
r
=−
v
r
+
p
=−
+
x
+
1   2 u  2 u 1 u
2
1  u
+
+
+


Re  x 2 r 2 r r r 2 2 
1   2 v  2 v 1 v 1  2 v v
2 w 
+
+
+
− −

Re  x 2 r 2 r r r 2 2 r 2 r 2
1 p
r 
1 w
r 
+
(2)

 

(3)
1   2w  2w 1 w 1  2w w 2 v 
+
+
+
− +

 (4)
Re  x 2 r 2 r r r 2 2 r 2 r 2  
=0
(5)
3.2 Boundary conditions
3. PROBLEM FORMULATION
The appropriate boundary conditions were considered in the
radial direction for the numerical solution of the eigenvalue
problem. On the surface of cylinder, no-slip and no-penetration
conditions were applied. So the magnitude of all velocity
perturbations are zero at the wall.
U(x, a) = 0, v(x, a) = 0, w(x, a) = 0
3.1 Geometry and Governing stability equations
A circular cylinder with different forebody shapes was
considered in the axial stream of the incompressible fluid. The
streamwise locations between the lines ab and dc were selected
for the stability analysis. The boundary layer development starts
from the stagnation point O, as shown in Fig.1, and the forebody
(bof) significantly affects the boundary layer developed on the
main body (cylinder).
In the radial direction, far away from the wall it is assumed that
all perturbations are going to decay. So at far-field, all velocity
and pressure perturbation approaches to zero magnitude.
U(x,) = 0, v(x,) = 0, w(x,) = 0, p(x,) = 0
3.3 Discretization
The discretization of the stability equations was done using the
Chebyshev Spectral Collocation method. The Chebyshev
polynomial generates a non-uniform grid and generates more
collocation points towards the end. It is a sufficient arrangement
for the boundary layer problems.
 j 
yc = cos   Where, j = 0, 1, 2, 3……….m
m
Figure 1: schematic diagram of the boundary layer on
a circular cylinder with the fore-body
2
(15)
Where m is the number of collocation points in wall-normal
direction. The following equation applies grid stretching:
yr =
yi L y (1 − yc )
L y + yc (L y − 2yi )
+a
A streamwise location with x = 1.21 m (from the leading edge
of a cylinder) and radial location, r = 0.011 m (from the
cylindrical surface), were selected to perform this test.
Table 1: Grid convergence study for the base flow
computations.
(16)
When all the partial derivatives are discretized by the
Chebyshev Spectral Collocation method, the equations (2) – (5)
can be written in the matrix form as given below.
A11
A21
A31
A41
A12
A22
A32
A42
A13
A23
A33
A43
A14
A24
A34
A44
u
v + 1
w
p
B11
B21
B31
B41
B12
B22
B32
B42
B13
B23
B33
B43
B14
B24
B34
B44
u
v
w
p
C11 C12 C13 C14
+ 2 C21 C22 C23 C24 = 0
Sr.
No
Grid size
U (m/s)
%
error
V (m/s)
%
error
#1
1001×125
0.0154920
---
2.5416
---
#2
1415×177
0.0154954
0.02190
2.5344
0.282
#3
2001×251
0.0154962
0.00516
2.5290
0.212
Table 1 shows the grid convergence test for the different grid
sizes. The grid size #3 has been selected for the base flow
computations in all results presented here. A SIMPLEC
algorithm with the second order upwind spatial discretization
scheme was used in all computations.
(17)
C31 C32 C33 C34
C41 C42 C43 C44
4.2 Baseflow profile for the different forebody shapes
 A +   B +  2 C  = 0
The baseflow profile is obtained for the different forebody
shapes within the boundary layer region to understand the effect
of forebody shapes on the profile. We have plotted the graph of
axial baseflow velocity ‘U’ versus radius ‘r’ (wall-normal
distance or y coordinate) at different streamwise locations. We
have considered the two different streamwise locations such
that first location is near the leading edge and the second
location is far away from the leading edge (or we can say
towards down-stream direction)
(18)
The equation (18) is a polynomial type eigenvalue problem with
the order of two. This equation was solved using the ‘ployeig’
MATLAB function. The above equation solution gives
complex eigenvalues as complex wave-number ( = r + ii)
If i > 0 the flow is stable, i = 0 the flow is neutrally stable,
and i < 0 the flow is unstable. Where, i is the spatial growth
rate of the disturbance.
4. BASEFLOW SOLUTION
4.1 Governing baseflow equations and
Grid convergence study
The baseflow solution was obtained using Finite Volume Code,
ANSYS Fluent. The steady Navier-Stokes equations were
solved in the axisymmetric domain with the three different
shapes of the fore bodies; ellipsoid, paraboloid, and sharp-cone.
The baseflow computations were done considering the domain
p-o-b-c-d-a-p, as shown in Figure 1. The governing baseflow
equations are given below:
U
U
x
+V
U
r
=−
P
x
+
1   2U  2U 1 U 


+
+
Re  x 2 r 2 r r 
V
V
P 1   2V 1 V  2V V 

U
+V
=−
+
+
+
− 
x
r
r Re  r 2 r r x 2 r 2 
V
r
+
V
r
+
U
x
=0
(19)
(a)
(b)
Figure 2: Baseflow velocity profile for different
forebody shapes for FR = 2.5 & Re = 1000 at
streamwise location (a) x = 0.9 (b) x = 50
(20)
The figure 2 (a) and (b) show the effect of forebody shapes on
the baseflow profile at the streamwise location x = 0.9 which is
near the leading edge and x = 50 which is towards the downstream direction. As shown in figure 2 (a), the forebody shapes
have significant effect on baseflow profile near the leading edge
but as shown in figure 2 (b), the forebody shapes have
negligible effect on the baseflow profile. Hence the effect of
forebody shapes on the baseflow profile is significant near the
leading edge only.
(21)
Where U and V are the base velocity components in the axial
and radial directions, appropriate boundary conditions were
applied at the inlet, outlet, wall, and free-stream for the
numerical solution of the N-S equations. Grid convergence tests
were done to check the proper grid size and accuracy of the
solution.
3
5. CODE VALIDATION
N = 0 and N = 2 respectively. The least stable eigenvalues are
marked by a square in both the eigenspectrum. The least stable
eigenvalues marked by square are C = 0.320 + 0i and C =.0.425
+ 0i are in close agreement with the results of Tutty et al.
(2002).
The spatial stability analysis has been performed as shown by
Eq. 18 to understand the effect of different shapes of fore-body
geometry on the stability of axisymmetric boundary layer. To
validate the stability computations, a blunt cylinder was
considered and temporal and spatial stability results were
validated against Tutty et al. (2002). First we performed
temporal stability analysis for axisymmetric (N = 0) and nonaxisymmetric mode (N = 2). The critical Reynolds number of
12439 (for N = 0) and 6070 (for N = 2) were considered [1]. As
we know that in temporal stability analysis, streamwise wavenumber  is real and known while complex wave frequencies
are computed as eigenvalues. However, in the spatial stability
analysis, wave-frequencies are real and known, and complex
wave-numbers are computed as eigenvalues.
Table 2: Results of temporal stability analysis of the
axisymmetric boundary layer [1].
Input parameters for temporal stability
Figure 5: Eigenspectrum for N = 0, Re = 12439,
streamwise location x = 47, and Cr = 0.317. The
complex eigenvalue marked by square is α = 2.71 +
0.0i.
Results
N
Re
Xc
c
Cr
0
12439
47
2.73
0.317
2
6070
91.1
0.775
0.422
Figure 6: Eigenspectrum for N = 2, Re=6070,
streamwise location x = 91.1, and Cr = 0.422. The
complex eigenvalue marked by square is α = 0.77 +
0.0i.
Figure 3: Eigenspectrum for N = 0, Re = 12439,
streamwise location x = 47, and  = 2.73. The complex
eigenvalue marked by square is C = 0.32 + 0.0i
In the second step, spatial stability analysis was performed with
obtained complex frequencies from the temporal stability
analysis as input (for critical Reynolds number imaginary part
of the eigenvalue is zero), and complex wave-numbers are
computed as eigenvalues. Figures 5 and 6 present the
eigenspectrum of complex wave-number resulting from the
spatial stability analysis. The wave-numbers (complex) in
Figures 5 and 6 are marked with a square. They have the almost
same magnitudes as 2.71 for N = 0 and 0.77 for N = 2 which
are in close agreement with the results of Tutty et al. (2002).
6. RESULTS & DISCUSSION
The axisymmetric boundary layer’s spatial stability analysis has
been performed on the modified baseflow due to the
axisymmetric forebodies mounted at the leading edge of the
circular cylinder. The ellipsoid, paraboloid, and sharp-cone
with FR = 2.5, 5, and 7.5 were considered at Re = 1000, 4000,
and 10000. The effect of fore-body shapes has been studied on
axisymmetric (N = 0) and non-axisymmetric modes (N = 2).
The grid convergence test was also performed for the stability
computations. The stability computations have been done for
the domain of cylinder only.
Figure 4: Eigenspectrum for N=2, Re = 6070,
streamwise location x = 91.1, and  = 0.775. The
complex eigenvalue marked by square is c= 0.425 +
0.0i
In the first step, a temporal stability analysis was performed for
N = 0, Re = 12439, streamwise location x = 47, and  = 2.73
and then for N = 2, Re = 6070, streamwise location x = 91.1,
and  = 0.775. Figure 3 and 4 show the eigenspectrum for the
4
6.1 Axisymmetric mode (N = 0)
favourable pressure gradient makes the boundary layer stable.
Figure 8 also indicates that the effect of FR on the spatial
growth rate is significant near the boundary layer's leading edge
while it gradually reduces in the streamwise direction towards
the downstream.
Figure 7: Variation of the spatial growth rate (-αi) for
Re = 12439, FR = 2.5, N = 0 for different axisymmetric
forebody shapes.
Figure 7 shows the variation of the spatial growth rate (-αi) of
the perturbations at the different streamwise locations for
axisymmetric mode, Re = 12439, and FR = 2.5. The comparison
shows that the spatial growth rate of the disturbances is highest
for the elliptical forebody and the least for the sharp cone
towards the leading edge. This indicates that the boundary layer
with the sharp-cone forebody is more stable, and the boundary
layer with the ellipsoid is less stable. At near the critical
location x = 47, the spatial growth rate is almost zero for all the
three forebody shapes. In the streamwise direction ahead of the
critical position, again, spatial growth starts to reduce, which
indicates that the boundary layer is going to relaminarize again.
The important observation is that the difference in the spatial
growth rate at different streamwise locations has been reduced
between all the three forebody shapes. It proves that the effect
of the forebody shapes on the stability of the boundary layer is
dominant near the leading edge only. The effect of these shapes
is almost the same away from the leading edge or towards the
downstream.
Figure 9: Variation of the spatial growth rate (-αi) in
the streamwise direction for paraboloid forebody for
N = 0, FR = 5, and ωr = 0.865 for different Re.
Figure 9 presents the spatial growth rate of the disturbances for
paraboloid forebody with FR = 5 and forcing frequency ωr =
0.865. The diagram shows that the spatial growth rate at any
particular streamwise location is highest for the Re=10000 and
lowest for the Re = 1000. It proves that increased Reynolds
number makes the flow locally unstable, and at critical
Reynolds number, it becomes fully unstable, and transition to
turbulence occurs.
6.2 Non-axisymmetric (Helical) mode
Azimuthal wave number N = 2 was selected at Critical
Reynolds number with FR = 2.5 to understand the effect of
forebody shapes on the axisymmetric boundary layer's helical
modes. Figure 10 shows the comparison of the spatial growth
rate for different forebody shapes at Re = 6070. The behaviour
for the helical mode N = 2 has been found qualitatively similar
to the axisymmetric mode.
Figure 8: Variation of the spatial growth rate (-αi) for
ellipsoid in the streamwise direction for N = 0, Re =
12439 for different FR.
Figure 8 shows the effect of different fineness ratio on the
spatial growth rate of the disturbances for ellipsoid forebody in
the streamwise direction. For a given forebody shape
(ellipsoid), the boundary layer is more stable with FR = 7.5 and
less stable for FR = 2.5. It evident that increased FR of the
forebody reduces the spatial growth rate of the disturbances,
and the boundary layer becomes more stable. The increased FR
increases the sharpness of the forebody, which develops a more
favourable pressure gradient. It is well established that a
Figure 10: Variation of the spatial growth rate (-αi) in
the streamwise direction for N = 2, Re = 6070, and FR
= 2.5 for different forebody shapes.
The x = 91.1 location is the critical point location for the helical
mode N = 2. It has been observed here that the spatial growth
rate is highest for the ellipsoid and the lowest for the sharp cone.
Similarly, the spatial growth rate difference is more significant
5
at the leading edge of the boundary layer and gradually reduces
in the streamwise direction. Beyond the critical point, again,
spatial growth rate reduces, thus boundary layer relaminarize
again towards the downstream.
The spatial growth rate for the ellipsoid forebody was highest,
while that of sharp-cone was found to be the least one for a
given Reynolds number and FR. The forebody with FR = 7.5
has the least spatial growth rate, while a forebody with FR = 2.5
has the highest spatial growth rate. The perturbation was found
spatially growing until the critical location and decayed when
Reynolds number smaller or equal to Critical values. The
perturbations were continuously growing in the streamwise
direction for Reynolds number larger than Critical value.
NOMENCLATURE
Re

α
ρ
ω
N
U, V, W
Figure 11: Variation of the spatial growth rate (-αi) in
the streamwise direction for ellipsoid for N = 2, Re =
6070, for different FR
Figure 11 shows the variation of the ellipsoid spatial growth
rate at Re = 6070 and N = 2. The spatial growth rate is found
the least for FR = 7.5.
u, v, w
U
a
Reynolds number
Viscosity of water
Streamwise wave-number
Density of water
Wave frequency
Azimuthal wave-number
Baseflow velocity in x, r and 
directions respectively
Disturbance velocity in x, r and 
directions respectively
Free stream velocity
Body radius of a cylinder
[m2]
Ns/m2
1/m
[kg/m3]
[rad/s]
1/m
m/s
m/s
m/s
m
REFERENCES
[1] O. R. Tutty and W. G. Price, Boundary layer flow on a long
thin cylinder, Physics of Fluid (2002)
[2] N. Vinod, H. Balakrishnan and R. Govindarajan, Stability
Analysis of an Axisymmetric Boundary Layer, Asian Congress
of Fluid Mechanics (2002)
[3] N. Vinod and R. Govindarajan, Linear and secondary
instabilities in incompressible axisymmetric boundary layers:
Effect of transverse curvature, J. Fluid Mech. (2006)
[4] M. B. Glauert and M. J. Lighthill, the Axisymmetric
Boundary Layer on a Long Thin Cylinder, the Royal Society
(1954)
[5] Y. Mérida, M. Jelliti and T. Lili, Laminar Instability of
Parallel and Nonparallel Flows of Adiabatic Flat Plate, Journal
of Applied Fluid Mechanics (2016)
[6] Z. Wan, H. Yang, L. Zhou, and D. Sun, Linear stability
analysis of supersonic axisymmetric jets, Theoretical and
Applied Mechanics Letters (2014)
[7] S. Muralidhar and R. Govindarajan, Linear stability analysis
of boundary layer over a long cylinder with rotation,
Conference: Congress Francais de Mecanique (2015)
[8] P. Kumar and K. Mahesh, Analysis of axisymmetric
boundary layers, Journal of Fluid Mechanics (2018)
[9] F. Liu, Y. WANG and Y. Piao, Linear stability analysis of
interactions between mixing layer and boundary layer flows,
Chinese Journal of Aeronautic (2016)
[10] G. Xin and T. Dengbin, Nonlinear Stability of Supersonic
Nonparallel Boundary Layer Flows, Chinese Journal of
Aeronautic (2009)
[11] M. Johnson and A. Pinarbasi, The Effect of Pressure
Gradient on Boundary Layer Receptivity, Flow Turbulence and
combustion (2014)
Figure 12: Variation of the spatial growth rate (-αi) in
the streamwise direction for paraboloid forebody
shape for N = 2, FR = 5, and ωr = 0.327 for different Re
Figure 12 shows the spatial growth rate of the disturbances for
different Reynolds numbers at forcing frequency of ωr = 0.327.
Re = 10000 is above the Critical value of the Reynolds number.
Thus, spatially flow is found unstable as well as in the
streamwise direction. Also, spatial growth increases towards
the downstream. It shows that the flow is unstable.
7. CONCLUSIONS
Local spatial stability analysis of the axisymmetric boundary
layer has been performed to study the effect of axisymmetric
forebody shapes on the stability of the boundary layer. We
considered ellipsoid, paraboloid, and sharp-cone geometry at
Re = 1000, 4000, and 10000 and spatial growth rate of the
perturbation was computed at different streamwise locations.
The polynomial eigenvalue problem was solved using the
ployeig function of the MATLAB.
It has been found that forebody shapes have a significant effect
on the base flow and stability characteristic near the leading
edge of the boundary layer. Towards the downstream
sufficiently away from the leading edge (x = 50), the spatial
growth rate for all three forebody shapes found almost the same.
6