See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/257378615
Numerical heat transfer study around a spiked blunt-nose body at Mach 6
Article in Heat and Mass Transfer · April 2013
DOI: 10.1007/s00231-012-1095-6
CITATIONS
READS
9
226
1 author:
R. C. Mehta
Noorul Islam University
102 PUBLICATIONS 569 CITATIONS
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Reentry aerodynamics, Venting, Inverse heat conduction problem View project
Drag Reduction for Spike Attached to Blunt-Nosed Body at Mach 6, Experimental investigation on spiked body in hypersonic flow View project
All content following this page was uploaded by R. C. Mehta on 23 September 2015.
The user has requested enhancement of the downloaded file.
Numerical heat transfer study around a
spiked blunt-nose body at Mach 6
R. C. Mehta
Heat and Mass Transfer
Wärme- und Stoffübertragung
ISSN 0947-7411
Heat Mass Transfer
DOI 10.1007/s00231-012-1095-6
1 23
Your article is protected by copyright and
all rights are held exclusively by SpringerVerlag Berlin Heidelberg. This e-offprint is
for personal use only and shall not be selfarchived in electronic repositories. If you
wish to self-archive your work, please use the
accepted author’s version for posting to your
own website or your institution’s repository.
You may further deposit the accepted author’s
version on a funder’s repository at a funder’s
request, provided it is not made publicly
available until 12 months after publication.
1 23
Author's personal copy
Heat Mass Transfer
DOI 10.1007/s00231-012-1095-6
ORIGINAL
Numerical heat transfer study around a spiked blunt-nose
body at Mach 6
R. C. Mehta
Received: 3 September 2011 / Accepted: 27 November 2012
Ó Springer-Verlag Berlin Heidelberg 2012
Abstract An aerospike attached to a blunt body significantly alters its flowfield and influences aerodynamic drag
at high speeds. The dynamic pressure in the recirculation
area is highly reduced and this leads to the decrease in the
aerodynamic drag. Consequently, the geometry of the
aerospike has to be simulated in order to obtain a large
conical recirculation region in front of the blunt body to get
beneficial drag reduction. Axisymmetric compressible
Navier–Stokes equations are solved using a finite volume
discretization in conjunction with a multistage Runge–
Kutta time stepping scheme. The effect of the various types
of aerospike configurations on the reduction of aerodynamic drag is evaluated numerically at a length to diameter
ratio of 0.5, at Mach 6 and at a zero angle of incidence. The
computed density contours are showing satisfactory
agreement with the schlieren pictures. The calculated
pressure distribution on the blunt body compares well with
the measured pressure data on the blunt body. Flowfield
features such as formation of shock waves, separation
region and reattachment point are examined for the flatdisc spike and on the hemispherical disc spike attached to
the blunt body. One of the critical heating areas is at the
stagnation point of a blunt body, where the incoming
hypersonic flow is brought to rest by a normal shock and
R. C. Mehta (&)
School of Mechanical and Aerospace Engineering,
Nanyang Technological University,
Singapore 639798, Singapore
e-mail: atulm@md4.vsnl.net.in
Present Address:
R. C. Mehta
Department of Aeronautical Engineering,
Noorul Islam University,
Kumaracoil 629180, India
adiabatic compression. Therefore, the problem of computing
the heat transfer rate near the stagnation point needs a
solution of the entire flowfield from the shock to the spike
body. The shock distance ahead of the hemisphere and the
flat-disc is compared with the analytical solution and a good
agreement is found between them. The influence of the
shock wave generated from the spike is used to analyze
the pressure distribution, the coefficient of skin friction and the
wall heat flux facing the spike surface to the flow direction.
List of symbols
Cf
Skin friction coefficient
CD
Aerodynamic drag
Cp
Pressure coefficient
D
Cylinder diameter
e
Specific energy
h
Enthalpy
H
Source vector
F, G Flux vector
k
Thermal conductivity
L
Length of the spike
M
Mach number
q
Wall heat flux
Pr
Prandtl number
p
Pressure
pa
Ambient pressure
s
Distance along the surface of the spike
T
Temperature
t
Time
u, v
Velocity components
W
Conservative vector
x, r
Coordinate direction
c
Ratio of specific heats
l
Viscosity
q
Density
123
Author's personal copy
Heat Mass Transfer
r
D
Stress vector
Shock stand-off distance
Subscripts
e
Boundary layer edge
F
Flat-disc
o
Stagnation
S
Hemispherical disc
w Wall
? Freestream condition
Fig. 1 Schematic sketch for flowfield over spiked blunt body
1 Introduction
A high-speed flow over a blunt body generates a bow shock
wave in front of it, which causes a rather high surface
pressure and, as a result, high aerodynamic drag. The
surface pressure on the blunt body can be substantially
reduced if a conical shock wave is generated by attaching a
forward-facing spike. Thus, the introduction of the spike
decreases the aerodynamic drag. The spike produces a
region of recirculating separated flow that shields the bluntnosed body from the incoming flow. It is advantageous to
have a vehicle with a low drag coefficient in order to
minimize the thrust required from the propulsive system
during the supersonic and hypersonic regime.
Many researchers have investigated flowfield simulations over the blunt body attached with forward facing
aero-spike. A typical flow over a spike attached to a blunt
body is based on experimental investigations. A schematic
of the flowfield over the aerodisc spiked blunt body at zero
angle of attack is shown in Fig. 1. The flow field around a
spiked blunt body appears to be very complex and contains
a number of interesting flow phenomena and characteristics, which are yet to be investigated. The recirculating
region is formed around the root of the spike up to
the reattachment point of the flow at the shoulder of the
hemispherical body. Due to the recirculating region, the
pressure at the stagnation region of the blunt body will
decrease. Flow behind the conical shock wave separates on
the spike and creates a conical shaped recirculation zone
appears in the vicinity of the stagnation region. Due to the
formation of the flow recirculation, the pressure and wall
heat flux are reduced in the forward facing region of the
blunt body. However, the reattachment of the shear layer
on the shoulder of the hemispherical body increases the
local heat flux and pressure. The reattachment shock is
moved downstream as depicted in the diagram, which is a
function of the geometrical parameter of the spike and the
shape of the spiked nose body.
The prime focus has been with regard to drag characteristics where effects due to spike length, spike head
geometry, forward body geometry and relative spike
123
diameter have been explored. Most of the experimental
studies on spiked bodies are carried out to get effects on
aerodynamic heating [1–3] and the surface pressure distributions [4–6] at supersonic and hypersonic Mach numbers [7–10]. Kubota [11] has experimentally investigated
the overall characteristics of the spiked blunt body configuration at hypersonic Mach numbers. Crawford [12]
experimentally investigated the effects of the spike length
on the nature of the flow field for a freestream Mach
number 6.8 and Reynolds number 0.12 9 106 - 1.5 9 106
based on the cylinder diameter.
The spike geometry drastically influences the aerodynamic drag of the blunted body at high-speeds. Motoyama
et al. [13] have experimentally investigated the aerodynamic and heat transfer characteristics of conical, hemispherical, flat-faced aerospikes, and hemispherical and
flat-faced discs attached to the aerospike for a freestream
Mach number 7, freestream Reynolds number 4 9 105/m,
for L/D = 0.5 and 1.0, and angle-of-attack 0 to 8°. They
found that the aerodisc spike (L/D = 1.0 and aerodisc
diameter of 10 mm) has a superior drag reduction capability as compared to the other aerospikes. Yamauchi et al.
[14] have numerically investigated the flowfield around a
spiked blunt body at freestream Mach numbers of 2.01,
4.14 and 6.80 for different ratio of L/D. Shoemaker [15],
Fujita and Kubota [16], Asif et al. [17] used a numerical
approach to solve the compressible Navier–Stokes equations. Their investigation shows that the reattachment point
can be moved backward or removed, which depends on the
spike length or the nose configuration. However, because
of the reattachment of the shear layer on the shoulder of the
hemispherical body, the pressure near that point becomes
large. Milicev et al. [18] have experimentally investigated
the influence of four different types of spikes attached to a
hemisphere-cylinder body at Mach number 1.89, Reynolds
number 0.38 9 106 based on the cylinder diameter, and at
an angle-of-attack of 2°. They observed in their experimental studies that a reliable estimation of the aerodynamic
effects of the spike can be made in conjunction with flow
visualization techniques. Numerical simulations [19–21]
have been carried out to obtain the comparative studies of
Author's personal copy
Heat Mass Transfer
the flowfield over the spike. Peak heating for the reattachment of the separated flow on a spiked blunt body is
numerically studied for freestream Mach numbers 2.01,
4.15 and 6.80. Recently, axisymmetric numerical simulations [22] have been performed for different types of spikes
attached to a blunt nose cone at Mach numbers of 5.0, 7.0
and 10.0 employing CFD-FASTRAN flow solver. It is
found in the numerical simulations that a spike with an
aerodisc outperformed the plain pointed aerospike in terms
of drag and aerodynamic heating reduction [23, 24].
Heat transfer rate on spike surface depends strongly upon
the stagnation point velocity gradient which is difficult to
estimate for the flat-nose spike. A flat nose spike reduces
stagnation point heat transfer rate [25] by up to 50 % as
compared to the hemispherical nose. The present paper presents a numerical simulation of the flowfield over a hemispherical and a flat-disc aerospike attached to a blunt body.
The focus of the present numerical analysis is to investigate
the mechanism of drag reduction and flowfield pattern at a
freestream Mach number 6. However, the aerodisc and aeroflat spiked blunt body include phenomena of flowfield facing
the flow direction that have not been described previously.
The flowfield features captured by the density, Mach and
pressure contours are used to understand the mechanism of
the drag reduction. The influence of the shock wave generated
from the spike is also studied to understand the pressure
distribution, the coefficient of skin friction and wall heat
transfer over the spike surface. The pressure ratio and the heat
transfer rate at the stagnation point are computed and compared with the analytical solution. The numerical results of
the research on a hemispherical and a flat-disc spike at L/D
ratio of 0.5 compared with the available experimental data are
found to be in good agreement.
H ¼ ½0; 0; rþ ; 0T
where the superscript T represents transpose matrix. The
viscous and heat flux terms in the above equations are
2
ou
rxx ¼ lr U þ 2l
3
ox
2
ov
rrr ¼ lr U þ 2l
3
or
ou ov
þ
rxr ¼ l
or ox
2
v
rþ ¼ p lr U þ 2l
3
r
ou ov v
þ þ
rU ¼
ox or r
oT
qx ¼ k
ox
oT
qr ¼ k
or
where CP is specific heat at constant pressure, U is the
mean stream velocity. l is calculated according to
Sutherland’s law. The flow is assumed to be laminar,
which is also consistent with Bogdonoff and Vas [1],
Yamauchi et al. [14], Fujita and Kubota [16] and Ahmed
and Qin [23]. T is related to p and q by the perfect gas
equation of state as
1
p ¼ ðc 1Þq e u2 þ v2
ð3Þ
2
The ratio of the specific heats is assumed constant and is
equal to 1.4.
2 Governing fluid dynamics equations
3 Numerical scheme
A numerical simulation is carried out to solve the unsteady,
compressible, axisymmetric Navier–Stokes equations over
a forward facing spike attached to a blunt body. The
governing equations can be written in the following strong
conservation form as
3.1 Spatial discretization
oW oF 1 oðrGÞ H
þ
þ
¼
ot
ox r or
r
where
ð1Þ
To facilitate the spatial discretization in the numerical
scheme, the time dependent axisymmetric compressible
Navier–Stokes equation (1) can be written in the integral
form over a finite volume as
Z
Z
Z
o
WdX þ ðFdr GdxÞ ¼ HdX
ð4Þ
ot
X
T
W ¼ ½q; qu; qv; qe
F ¼ qu; qu2 þ p rxx ; quv rxr ; ðqe þ pÞu
urxx vrxr þ qx T
G ¼ qv; quv rxr ; qv2 þ p rrr ; ðqe þ pÞv
T
vrrr urxr þ qy
ð2Þ
C
X
where X is the computational domain, C is the boundary
domain. The contour integration around the boundary of
the cell is taken in the anticlockwise sense. The numerical
technique uses a finite volume method on a structured nonoverlapping quadrilateral mesh. The spatial and temporal
terms are decoupled using the method of lines. The flux
vector is divided into the inviscid and viscous components.
123
Author's personal copy
Heat Mass Transfer
A cell-centered scheme is used to store the flow variables.
The discretization of inviscid fluxes is performed using
the cell average scheme. The discretization of viscous
fluxes involves the centre difference scheme of the second
order. Thus, the discretized solution to the governing
equations results in a set of volume-averaged state variables of mass, momentum, and energy which are in balance with their area-averaged fluxes (inviscid and viscous)
across the cell faces [26]. The finite volume code constructed in this manner reduces to a central difference
scheme and is second-order accurate in space provided
that the mesh is smooth enough [27]. The cell-centered
spatial discretization scheme is non-dissipative; therefore,
artificial dissipation terms are included as a blend of a
Laplacian and biharmonic operator in a manner analogous
to the second and fourth difference. The artificial dissipation term [28] was added explicitly to prevent numerical
oscillations near the shock waves to damp high-frequency
modes.
3.2 Time marching scheme
Temporal integration was performed using the three-stage
time-stepping scheme of Jameson et al. [28] based on the
Runge–Kutta method. The artificial dissipation is evaluated
only in the first stage. The details of this flowfield technique are further described in Ref. [29]. The solver uses a
time-marching procedure to compute the flow. The flow is
defined to be steady because the flowfield is converging to
a steady state. The steady options use local time stepping,
which leads to a faster convergence to the steady-state flow
field.
3.3 Initial and boundary conditions
Conditions corresponding to a freestream Mach number
6.0 were given as initial conditions. All the variables
were extrapolated at the outer boundary, and the no-slip
wall condition was used to implement the boundary
conditions. An isothermal wall condition was prescribed
for the surface of the model, that is, a wall temperature
of 300 K. The symmetric condition was applied on the
centerline.
4 Model and grid generation
4.1 Spike geometry
The dimensions of the spiked blunt body considered in the
present analysis are depicted in Fig. 2. The model is axisymmetric, the main body has a hemispherical-cylinder
123
Fig. 2 Dimensions of the spiked blunt body. a Flat-disc aerospike,
b hemispherical disc aerospike
nose, and diameter D is 4.0 9 10-2 m. The spike consists
of an aerodisc part and a cylindrical part. The hemispherical cap of the spike has diameter 0.2 D attached to a stem
of diameter of 0.1 D and L = 1.5 D as shown in Fig. 2a.
The flat-disc has an identical dimension of the hemispherical spike as shown in Fig. 2b.
4.2 Computational grid
One of the controlling factors for the numerical simulation
is the proper grid arrangement. The following procedure is
used to generate grid in the computational region of the
spiked blunt body. The computational domain is divided
into a number of non-overlapping zones. The mesh points
are generated in each zone using finite element methods
[30] in conjunction with the homotopy scheme [31]. The
spiked blunt nosed body is defined by a number of grid
points in the cylindrical coordinate system. Using these
surface points as the reference nodes, the normal coordinate is then described by the exponentially stretched grid
points extending outwards up to an outer computational
boundary. Grid independence tests [32] were carried out,
taking into consideration the effect of the computational
domain, the stretching factor to control the grid intensity
near the wall, and the number of grid points in the axial
and normal directions. The outer boundary of the computational domain was varied from 2.5 to 3.0 times the
cylinder diameter D and the grid-stretching factor in the
radial direction is varied from 1.5 to 5. These stretched
grids were generated in an orderly manner. To verify the
chosen grid delivers an accurate solution, the number of
grid cells was increased until a steady state solution
Author's personal copy
Heat Mass Transfer
Fig. 3 Enlarged view of computational grid. a Hemispherical disc aerospike, b flat-disc aerospike
occurred, that is, the resulting axial force on the investigated shape did not change anymore. Several test runs
were made with a total doubled grid cell number.
Therefore, the grid was highly refined in both directions.
Grids were chosen with the number of grid points in the
axial direction ranging from 187 for the shortest blunt
spike to up to 220 for the longest spike configuration, and
the number in the radial direction ranging from 52 to 82.
The present numerical analysis was performed on
187 9 62 grid points. The downstream boundary of the
computational domain is maintained at 4–6 times the
cylinder diameter. This grid arrangement was found to
give a relative difference of about ±1.5 % for the drag
coefficient. A convergence criterion of 10-5 is used, based
on the difference in the density values at any grid point
between two successive iterations. The minimum spacing
for the fine mesh is dependent upon the Reynolds number.
The finer mesh near the wall helps to resolve the viscous
effects. The coarse-mesh helps reducing the computer
time. A close-up view of the computational grid over the
hemispherical and the flat-faced aerospike is shown in
Fig. 3. The structured grid generation and the mono-block
are suitable to accommodate the spike shape. As seen in
the figures, these types of grid use quadrilateral cells in
2-D in the computational array. The quadrilateral cells,
which are very efficient at filling space, support a high
amount of skew and stretching before the solution will be
significantly affected. Additionally, the grid can be
aligned with the flow, thereby yielding greater accuracy
within the solver. Several grid arrangements are considered to verify the grid independency. The numerical
results are validated with the available experimental data
in the next section.
5 Results and discussion
5.1 Flowfield visualization and characteristics
Characteristic features of the flowfield around the hemispherical and the flat-disc aerospike attached to blunt body
at high speeds were investigated with the help of velocity
vector, density, pressure and Mach contours plots. Figure 4
depicts the velocity vector plots over the hemispherical
aerospike and the flat-disc aerospike for L/D = 0.5 at
M? = 6 and Reynolds number 0.979 9 107 based on the
cylinder diameter. The spiked body is completely enveloped within the recirculation region. The bow shock
interacts with the reattachment shock generated by the
blunt body. The interaction of the shock wave produced by
the hemispherical aerospike differs significantly with the
flat disc spike. The flow separation on the spike and
recirculation zone formed on the blunt body cap depends
on the shape of the spike. The contour plots explain the
cause of the drag reduction due to increase of the separation region over the spike. Figure 5 shows the enlarged
view of the nondimensional pressure contour plots over the
hemispherical and the flat disc spike attached to the blunt
body. It can be observed from the pressure contour plots
low and high pressure region over the spike. Figure 6
depicts the close-up view of the Mach contour plots over
the spiked-blunt body. The contour plots will help to locate
the high Mach zones. Figure 7 displays the zoomed region
of the vector plot on the hemispherical and flat-disc spike
configurations. The bow shock wave follows the aerospike
contour and the fore body is entirely subsonic up to the
corner tangency point of the flat-faced and the hemispherical aerospike where the sonic line is located. The
123
Author's personal copy
Heat Mass Transfer
Fig. 4 Velocity vector plot
over spiked-blunt body.
a Hemispherical disc aerospike,
b flat-disc aerospike
effects of the subsonic flow on the hemispherical and the
flat disc bodies have been investigated by Truitt [33].
5.2 Shock stand-off distance
The computed density contour plots with schlieren pictures [34] are shown in Fig. 8. The separated shear layer
and the recompression shock from the reattachment point
on the shoulder of the hemispherical body are visible in
the contour plot. The shock wave in front of the spike
cap will reduce the drag as compared to the case without
the spike. In the fore region of the aerodisc, the fluid
decelerates through the bow shock wave. At the shoulder
of the aerodisc or hemispherical cap, the flow turns and
expands rapidly, the boundary layer detaches, forming a
free shear layer that separates the inner recirculating flow
region behind the base from the outer flowfield. The
corner expansion over aerodisc process is a Prandtl–
Meyer distorted by the presence of the approaching
boundary layer. The computed flowfields show good
agreement with the schlieren photographs. The flowfields
are very different between the hemispherical and the flat-
123
faced spike as seen in the contour plots and are presented using schematic sketch in Fig. 8. For the case of
a flat-nosed spike flying at hypersonic speeds, a detached
bow wave is formed in front of the nose which is
practically normal to the body axis [33]. Since the flow
behind the normal shock is always subsonic, simple
continuity considerations show that the shock-detachment
distance and stagnation-velocity gradient are essentially a
function of the density ratio across the shock. The flow
behind the shock wave is subsonic, the shock is no
longer independent of the far-downstream conditions. A
change of the spike shape (geometry) in the subsonic
region affects the complete flow field up to the shock.
Figure 8 also represent schematic shock stand-off distance and the location of the sonic line. The shockdetachment distance becomes smaller with increasing
density ratio. Probstein [35] gives an expression for the
shock detachment distance DF (Fig. 8a) with diameter of
the flat-disc DS ratio as
rffiffiffiffiffiffiffi
DF
q1
¼ 2:8
ð5Þ
DS
q0
Author's personal copy
Heat Mass Transfer
Fig. 5 Enlarged view of
pressure contours.
a Hemispherical disc aerospike,
b flat-disc aerospike
Fig. 6 Enlarged view of Mach
contours. a Hemispherical disc
aerospike, b flat-disc aerospike
where the density ratio across the normal shock [36] is
e¼
2
1ÞM1
q1 ðc þ2
¼
2
ðc þ 1ÞM1
q0
ð6Þ
The ratio of shock stand-off distance DS with hemispherical
spike of diameter, DS (Fig. 8b) is
DS
2e
qffiffiffi
¼
DS 1 þ 8e
3
ð7Þ
The values of DF/DS and DS/DS are found to be 0.1898 and
0.1109, respectively. The numerical values of the ratio of
shock stand-off to spike cap diameter are calculated from
the velocity vector and pressure contour plot and they are
0.19 and 0.11 which show good agreement with the analytical values. The spherical spike shows the greatest
change in velocity gradient as compared to the flat disc.
The shock wave stands in front of the blunt body and forms
a region of subsonic flow around the stagnation region.
123
Author's personal copy
Heat Mass Transfer
Fig. 7 Close-up view of
velocity vector plot over spikedblunt body. a Hemispherical
disc aerospike, b flat-disc
aerospike
5.3 Surface pressure distribution
The pressure coefficient [Cp = 2{(p/p?) - 1}/cM2?]
variation on the blunt-nosed body with different spike
configurations are shown in Fig. 9. The computed pressure
123
distributions compare well with the experimental pressure
measurement data [34]. The x/R = 0 location is the spike/
nose tip junction, where R is radius of the cylinder. The
location of the maximum pressure on the surface of
the spiked blunt body is at a body angle of about 40°.
Author's personal copy
Heat Mass Transfer
Fig. 8 Flowfield for spiked blunt body and schematic shock location. a Density contour plot and schlieren picture for hemispherical disc
aerospike, b density contour plot and schlieren picture for flat-disc aerospike
The location corresponds to the reattachment point. It is
interesting to note that the maximum pressure is found on
the same location on the blunt body. The low pressure
ahead of the blunt body shows the cause of the reduction in
the aerodynamic drag.
Figures 10, 11, 12 depict the variation of non-dimensional pressure p/pa, skin friction coefficient and heat flux
over the spike surface facing the flow direction along the
spike. The s/DS = 0 is the location of the stagnation point.
DS is diameter of the spike as shown in Fig. 8. The sonic
line position on the flat and the hemispherical spike disc are
calculated using the computed pressure, isentropic and
normal shock relations [33] and are found to be 0.095 and
0.1, respectively. It shows that the sonic line appears on the
shoulder of the hemispherical spike. The pressure ratio p/pa
on the stagnation point is 48.84 and 38.23 for the flat-disc
and the hemispherical disc spike, respectively. The pressure ratio across the normal shock [37] is 41.83. It shows
123
Author's personal copy
Heat Mass Transfer
2.00
1.5x10
7
1.0x10
7
0.5x10
7
Experimental flat disk spike
Experimental hemispherical spike
Numerical hemispherical spike
Numerical flat-disk spike
1.75
2
qw (W/m )
1.50
Cp
1.25
1.00
0.75
Flat-disc spike
Hemispherical spike
0.50
0.25
0
0
0
0.5
1.0
1.5
2.0
2.5
3.0
0
0.02
0.04
0.06
0.08
0.10
s/DS
3.5
x/R
Fig. 12 Heat flux variation on the spike facing the flow direction
Fig. 9 Pressure distribution along the spike attached to blunt nosed
body
Table 1 Calculated aerodynamic drag for L/D = 0.5
50
Spike geometry
CD
Hemispherical spike
0.576
Flat face spike
0.458
40
Flat-disc spike
Hemispherical spike
p/pa
30
20
10
0
0
0.02
0.04
0.06
0.08
0.10
s/DS
Fig. 10 Pressure distribution on the spike facing the flow direction
-4
1.25x10
-4
Flat-disc spike
Hemispherical spike
1.00x10
-4
Cf
0.75x10
-4
0.50x10
-4
0.25x10
0
0
0.02
0.04
0.06
0.08
0.10
s/DS
Fig. 11 Skin friction variation on the spike facing the flow direction
that the percentage pressure ratio difference of the order of
16.77 and -8.59 % for the flat-disc and the hemispherical
disc spike, respectively. The difference is attributed to the
123
finite compressibility in the shock and the spike surface.
The aerodynamic drag is given in Table 1.
Newtonian flow [33] and similarity method [38] use the
given shock radius, freestream velocity, and density ratio
across the shock to determine the body surface where the
normal component of the velocity vanishes. These methods
do not take into consideration the finite compressibility that
exists between the shock wave and the spike surface. The
numerical analysis is able to take into consideration the
compressibility effects in the subsonic region.
5.4 Wall heat flux
The inviscid flow field in the vicinity of the stagnation point is
described in a fluid dynamics sense as the conversion of a
unidirectional high velocity stream by a normal shock wave
into a high temperature subsonic layer, which is taken to be
inviscid and incompressible [35]. At the stagnation point of a
blunt body, the incoming hypersonic flow brought to rest by a
normal shock and adiabatic compression. The heat transfer
rate is directly proportional to the enthalpy gradient at the
wall and square root of the tangential velocity gradient at
the edge of the boundary layer. The inviscid flow field in the
vicinity of the stagnation point is described as the conversion
of the unidirectional, high velocity stream by a normal shock
wave into a high temperature subsonic layer. The enthalpy
gradient is determined by the shape of the velocity profile in
the boundary layer and by the variation of the air properties
with temperature. Heat flux at the stagnation point can
be calculated using the following expression of Fay and
Riddell [39]
Author's personal copy
Heat Mass Transfer
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
qw lw
ðqe le K Þ
ð he h w Þ
qe le
0:6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qw ¼ 0:763 Pr
ð8Þ
The value of stagnation point velocity gradient K is taken
as 0.3 [25]. The hemispherical spike shows the greatest
changes in the velocity gradient as compared to the flat disc
spike. The magnitude of the stagnation-velocity gradient
indicates the maximum heat transfer over the hemispherical spike. The computed values of the stagnation point heat
flux are 0. 556 9 107 and 1.45 9 107 W/m2 for the flatdisc and the hemispherical disc spike, respectively. The
stagnation point calculated using Eq. (8) is 0. 831 9 107
and 1.55 9 107 W/m2 for the flat-disc and the hemispherical disc spike, respectively. The discrepancy is due to
the value of K. The value of K is difficult to calculate for
the flat disc spike [25]. The results show reasonably good
agreement between them. The close-up view of velocity
vector plot over the spiked-blunt body gives a comparative
velocity gradient in the s direction of the spike. The
velocity vector is turning in the stream line fashion on the
hemispherical spike whereas on the flat-disk spike it
appears the flow is impinging.
6 Conclusion
The flowfield around a forward facing hemispherical and
flat disc spike attached to blunt body has been numerically
simulated at a freestream Mach number 6, at length to
diameter ratio of 0.5 and at zero angle of attack. The flow
visualizations were done using the velocity vector and
contour plots in order to analyze the influence of the shape
of the spike on the drag reduction. The computed contour
plots agree well with the schlieren pictures of the experiments and the computed pressure distribution compares
well with measured pressure on the blunt body. The bow
shock wave formation is found over the spherical and the
flat spike which generate different separation zones over
the blunt body. The shock wave stand-off distances for the
hemispherical and the flat spiked are compared with the
analytical results and found in good agreement. The pressure distribution, the coefficient of skin friction and the
wall heat flux variation along the surface of the spike
facing the flow direction are influenced by the spike shape.
The pressure and the heat transfer rate at the stagnation
point are computed and compared with the analytical
solution. The numerical analysis gives complete flowfield
information over the spike surface including the shock
stand-off distance, sonic line, and velocity gradient along
the surface of the spike. The hemispherical disc spike gives
high aerodynamic drag and heat flux as compared to the
flat-faced disc spike.
Acknowledgments The author expresses his sincere gratitude to the
Editor and Referees for giving their valuable comments, suggestions,
and encouragement towards the improvement of the present work.
References
1. Bogdonoff SM, Vas IE (1959) Preliminary investigations of spike
bodies at hypersonic speeds. J Aerosp Sci 26(2):65–74
2. Maull DJ (1960) Hypersonic flow over symmetric spiked bodies.
J Fluid Mech 8:584
3. Wood CJ (1961) Hypersonic flow over spiked cones. J Fluid
Mech 12:614
4. Menezes V, Saravanan S, Jagdeesh G, Reddy KP (2003) Experimental investigation of hypersonic flow over highly blunted
cones with aerospikes. AIAA J 41(10):1955–1966
5. Mehta RC (2000) Pressure oscillations over a spiked blunt body
at hypersonic Mach number. Comput Fluid Dyn J 9(2):88–95
6. Mehta RC (2002) Numerical analysis of pressure oscillations
over axisymmetric spiked blunt bodies at Mach 6.8. Shock Waves
11:43–440
7. Milicv SS, Pavlovic MD, Ristic S, Vitic A (2001) On the influence of spike shape at supersonic flow past blunt bodies. Fac
Univ Ser Mech Autom Control Robot 3(12):371–382
8. Calarese W, Hankey WL (1985) Modes of shock wave oscillations on spike tipped bodies. AIAA J 23(2):185–192
9. Hahn M (1966) Pressure distribution and mass injection effects in
the transitional separated flow over a spiked body at supersonic
speed. J Fluid Mech 24:209–223
10. Milicev SS, Pavlovic MD (2002) Influence of spike shape at
supersonic flow past blunt nosed bodies experimental study.
AIAA J 40(5):1018–1020
11. Kubota H (2004) Some aerodynamic and aerothermodynamic
considerations for reusable launch vehicles. AIAA 2428
12. Crawford DH (1959) Investigation of the flow over a spiked-nose
hemisphere at a Mach number of 6.8. NASA TN-D 118
13. Motoyama N, Mihara K, Miyajima R, Watanuki W, Kubota H
(2001) Thermal protection and drag reduction with use of spike in
hypersonic flow. AIAA paper 2001-1828
14. Yamauchi M, Fujjii K, Tamura Y, Higashino F (1993) Numerical
investigation of hypersonic flow around a spiked blunt body.
AIAA paper 93-0887
15. Shoemaker JM (1990) Aerodynamic spike flowfields computed to
select optimum configuration at Mach 2.5 with experimental
validation. AIAA paper 90-0414
16. Fujita M, Kubota H (1992) Numerical simulation of flowfield
over a spiked blunt nose. Comput Fluid Dyn J 1(2):187–195
17. Asif M, Zahir S, Kamran N, Kan MA (2004) Computational
investigations aerodynamic forces at supersonic/hypersonic flow
past a blunt body with various forward facing spikes. AIAA paper
2004-5189
18. Milicev SS, Pavlovic MD, Ristic S, Vitic A (2002) On the
influence of spike shape at supersonic flow past blunt bodies.
Mech Autom Control Roboti 3(12):371–382
19. Mehta RC (2000) Peak heating for reattachment of separated flow
on a spiked blunt body. Heat Mass Transf 36:277–283
20. Mehta RC, Jayachandran T (1997) Navier-stokes solution for a
heat shield with and without a forward facing spike. Comput
Fluids 26(7):741–754
21. Mehta RC (2000) Heat transfer study of high speed over a spiked
blunt body. Int J Numer Meth Heat Fluid Flow 10(7):750–769
22. Gauer M, Paull A (2008) Numerical investigation of a spiked
nose cone at hypersonic speeds. J Spacecr Rocket 45(3):459–471
23. Ahmed MYM, Qin A (2010) Drag reduction using aerodisks for
hypersonic hemispherical bodies. J Spacecr Rocket 47(1):62–80
123
Author's personal copy
Heat Mass Transfer
24. Mehta RC (2010) Numerical simulation of the flow field over
conical, disc and flat spiked body at Mach 6. Aeronaut J
114(1154):225–236
25. White FM (1991) Viscous fluid flow, 2nd edn. McGraw Hill
International Edition, Singapore
26. Peyret R, Vivind H (1993) Computational methods for fluid
flows. Springer, Berlin, pp 109–111
27. Blazek J (2001) Computational fluid dynamics: principles and
applications, 1st edn. Elsevier Science Ltd, Oxford
28. Jameson A, Schmidt W, Turkel E (1981) Numerical simulation of
euler equations by finite volume methods using Runge–Kutta
time stepping schemes. AIAA paper 81-1259
29. Mehta RC (1998) Numerical investigation of viscous flow over a
hemisphere-cylinder. Acta Mech 128(1–2):48–58
30. Mehta RC (2011) Block structured finite element grid generation
method. Comput Fluid Dyn J 18(2):140–144
31. Shang JS (1984) Numerical simulation of wing-fuselage aerodynamic interference. AIAA J 22(10):1345–1353
32. Mehta RC (2000) Numerical heat transfer study over spiked-blunt
body at Mach 6.8. AIAA paper 2000-0344
123
View publication stats
33. Truitt RW (1959) Hypersonic aerodynamic. Ronald Press Co.,
New York
34. Kalimuthu R (2009) Experimental investigation of hemispherical
nosed cylinder with and without spike in a hypersonic flow. Ph.
D. thesis, Department of Aerospace Engineering, Indian Institute
of Technology, Kanpur, India, April 2009
35. Probstein RF (1956) Inviscid flow in the stagnation region of very
blunt-nosed bodies at hypersonic flight speeds. WADC–TN
56-395, Sept 1956
36. Ames Research Staff (1953) Equations, tables and charts for
compressible flow, NACA report 1135
37. Liepmann HW, Roshko A (2007) Elements of gas dynamics, 1st
South Asian Edition. Dover Publications Inc, New Delhi
38. Hayer WD, Probstein RF (1959) Hypersonic flow theory. Academic Press, New York
39. Fay JA, Riddell FR (1958) Theory of stagnation point heat
transfer in dissociated air. J Aeronaut Sci 25:73–85