16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
Validation of Shock-Expansion Tool for Two-dimensional and
Axisymmetric Flows with CFD
Jyothi Kumar Puttam
Shankara Rao Gonti1
Computational & Theoretical Fluid Dynamics Division
Council of Scientific and Industrial Research - National Aerospace Laboratories
Bangalore - 560017, INDIA.
Corresponding author e-mail: jyothi@ctfd.cmmacs.ernet.in
Abstract
The paper discusses the development and testing of shock-expansion tool for 2-D and axisymmetric supersonic and hypersonic flows.
The motivation is to compute the first-cut inviscid surface flow parameters on bodies with sharp leading edges and attached shocks
in minimum possible time without the necessity of grid. The flow parameters on the body surface immediately behind the leading
edge shock are computed using either by solving oblique shock relation(two-dimensional flows: plane shock case) or by solving TaylorMaccoll equation (axisymmetric flows: conical shock case). The subsequent flow parameters are computed either by Prandtl-Meyer
relation (for isentropic expansion) or again by oblique shock relation (for compression) depending upon local surface inclination. The
transcendental equations come into picture, while solving oblique shock or Prandtl-Meyer relations are solved using bi-section method.
The Taylor-Maccoll equation is a second order ODE and was solved using RK-4 method. In the present study shock-expansion tool
was tested for three 2-D bodies at different angle of attacks and three axisymmetric bodies at different Mach numbers with zero
angle of incidence. In almost all the cases the results are in good agreement with Euler results computed using in-house RANS solver
MB-EURANIUM and predicted the trends of experimental ones.
Key words: Supersonic Flow, Hypersonic Flow, Inviscid analysis, Oblique shock, Prandtl-Meyer expansion, Bisection method.
1 Introduction
In the very early stage of design cycle usage of some analytical tools helps in saving lot of time by
filtering out some designs, which are unable meet the specification requirements, and thus eliminates
detailed analysis on those. In supersonic and hypersonic vehicles design, inviscid flow analysis is one
such option, which eliminates the computationally expensive and much time consuming viscous flow
analysis, in the very early stage. For inviscid analysis of supersonic/hypersonic aerodynamic flows,
shock-expansion method is one such tool, which falls in the class of local surface inclination methods,
and computes the surface flow parameters [1]. In supersonic/hypersonic flows there are three families of
characteristics, viz the outward-running Mach waves from the body surface, the inward-running Mach
waves from the shock and stream lines. The shock-expansion method neglects the interaction of first
family of characteristics with last two families and thus gives the surface flow parameters exactly, only
when the neglected interactions are really negligible [2].
The surface flow parameters over two-dimensional or axisymmetric bodies with sharp leading edges
and attached shocks can be predicted with shock-expansion method. The first step of shock-expansion
method is to find the flow properties immediately behind the nose shock. For two-dimensional bodies(plane shock case), it can be accomplished by solving oblique shock relation, where as for axisymmetric bodies(conical shock case) the solution of Taylor-Maccoll equation is required. The subsequent
flow parameters are computed either by Prandtl-Meyer relation (for isentropic expansion) or again by
1
Project Student, Department of Aerospace Engineering, HAA, Marathahalli, Bangalore
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
oblique shock relation (for compression) depending upon local surface inclination.
The concept of shock-expansion theory was first used by Epstein [3] for determining the surface pressure
of an airfoil behind a shock. This concept of surface pressure determination was extended by Eggers et al
[4] for approximate determination of shock shape and entire flow field as ”generalised shock-expansion
method”. Later Syvertson et al [5] extended generalised shock-expansion method to bodies of revolutions near zero lift as second-order shock-expansion method. Jackson et al [6] combined second-order
shock-expansion theory with modified Newtonian theory to compute flow field on blunt nosed configuration with/without flares. Later DeJarnette et al [7] made lot of improvements to find exact pressure
gradient expression downstream a corner and found a new technique to find surface pressures at different
angle of incidence. Later real gas effects were also included with second-order shock-expansion method
by Moore et al [8].
The aim of the present work is to develop and test shock-expansion tool. First it was tested on three
2-D bodies viz, 5% thick diamond airfoil, 10% thick bi-convex airfoil and center-line symmetric section
contour of all-body hypersonic aircraft at different angle of attacks. Next the tool was tested on three
axisymmetric bodies viz, cones with different semi-vertex angles, tangent-ogive cylinder with fineness
ration of 3 and cone-cylinder-flare configuration at zero angle of incidence for different Mach numbers.
In all the cases the results were compared with Euler results computed using in-house RANS solver MBEURANIUM and available experimental results.
2 Shock-expansion tool: solution methodology
As said in introduction the flow should be supersonic/hypersonic on the entire surface to utilize shockexpansion method and the body should have a sharp pointed nose to hold an attached oblique shock either
plane or conical one. Shock-expansion method consists of two steps. The first step is to determine the
flow parameters immediately behind the nose shock either by solving oblique shock relation for plane
shock case or by solving Taylor-Maccoll equation for conical shock case. The second step is to determine
the subsequent flow parameters on entire surface either by solving Prandtl-Meyer relation for isentropic
expansion fan case or by solving oblique shock relation for compression shock case, depending upon
local surface inclination. The solution techniques used in the present work for solving each of the above
equations is explained in the following sections.
2.1
Oblique shock relation
For a given free-stream Mach number, M the flow deflection angle, θ and oblique shock angle, β can be
related with oblique shock relation as follows:
tan θ = 2 cot β
M 2 sin2 β − 1
. . . . . . . . . (1)
M 2 (γ + cos 2β) + 2
Equation 1 is called as θ-β-M relation, which is a transcendental equations and was solved using bisection method.
2.2
Taylor-Maccoll equation
Inviscid conical supersonic flows, with the assumption of constant parameters along rays from apex, are
governed by Taylor-Maccoll equation, which is a second order ODE and was solved using Range-Kutta
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
4th order method through an inverse approach. In contrast to finding the shock angle for a given cone
(direct approach), we assume a shock angle and find the semi-vertex angle of cone, which supports that
shock. By iterating on shock angle with bi-section method, the shock angle of the given right circular
cone can be approached.
2.3
Prandtl-Meyer relation
The isentropic expansion of supersonic flows are governed by Prandtl-Meyer relation, which is also a
transcendental equations and was solved like oblique shock relation.
3 Results
3.1
Two-dimensional flows
In all two-dimensional test cases 200 equidistant points were considered on both upper and lower surface
for shock-expansion tool, where as 400 X 50 cells O-grids were used for inviscid CFD analysis.
3.1.1 5% thick diamond airfoil
The tests were carried out at a free-stream Mach number of 6.86 and at 0◦ , 5◦ , 10◦ , 15◦ and 20◦ angle
of attacks. Figure 1(a) shows the Euler grid used for inviscid CFD analysis, where as figure 1(b) shows
pressure contours with streamlines around diamond airfoil at 20◦ angle of attack. The experimental
results of McLellan et al [9] were used for comparison, which were conducted at Langley 11-inch hypersonic tunnel at Reynolds number of 0.98X106 . Figure 2 shows the comparison of pressure coefficient
((P − P∞ )/P∞ ) between Shock-expansion, experimental and Euler CFD results at different angle of
attacks. It can be seen that there is good agreement between shock-expansion and Euler CFD results.
But they over predicted compared to experimental results on suction side and the deviation increased
with angle of attack, which can be attributed to increasing viscous effects with angle of attack, where as
there is very good agreement on pressure side between all three sets of results.
3.1.2 10% thick bi-convex airfoil
The tests were carried out at a free-stream Mach number of 6.86 and at 0◦ , 6◦ , 10◦ , 15◦ and 20◦ angle of
attacks. Figure 3(a) shows the Euler grid used for inviscid CFD analysis and figure 3(b) shows pressure
contours with streamlines around bi-convex airfoil at 20◦ angle of attack. The experimental results of
McLellan et al [9] were used for comparison, which were conducted at Langley 11-inch hypersonic
tunnel at Reynolds number of 0.98X106 . Figure 4 shows the comparison of pressure coefficient ((P −
P∞ )/P∞ ) between Shock-expansion, experimental and Euler CFD results at different angle of attacks.
Bi-convex airfoil results have exhibited very much similar behaviour of diamond airfoil.
3.1.3 All-body hypersonic aircraft pitch plane contour
All-body hypersonic aircraft model is representative of a hypersonic cruise vehicle, which was used to
establish benchmark experimental data base for validation of advanced CFD codes. The experiments
were conducted at NASA Ames 3.5-foot Hypersonic Wind Tunnel by Lockman et al [10].In the present
study, the pitch plane symmetric section surface contour of all-body hypersonic aircraft was used as a
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
(P
- P ∞) / P ∞
P/P_inf
13
12
11
10
9
8
7
6
5
4
3
2
1
(a) Euler grid around diamond airfoil
(b) Pressure contours with streamlines at aoa = 20◦
Figure 1: Euler grid and pressure contours on 5% thick diamond airfoil at free-stream Mach number of
6.86 and 20◦ angle of attack
2-D test case for shock-expansion tool, to test at a free-stream Mach number of 7.4 and at 0◦ , 5◦ , 10◦
and 15◦ angle of attacks. Figure 5(a) shows the Euler grid used for inviscid CFD analysis and figure 5(b)
shows pressure contours with streamlines around all-body hypersonic aircraft pitch plane contour at 15◦
angle of attack. The same experimental results of Lockman et al [10] were used for comparison, which
were conducted at Reynolds number of 15X106 . Figures 5 (c) and (d) shows the comparison of pressure
coefficient (P/P∞ ) between shock-expansion, experimental and Euler CFD results at different angle of
attacks respectively on leeward and windward sides of all-body hypersonic aircraft pitch plane contour.
Like in the previous cases shock-expansion and Euler CFD results are in very good agreement on both
suction and pressure sides, where as they underpredicted on suction side. On pressure side inviscid
results overpredicted from experimental ones and the deviation increased with angle of attack, but within
9% band at maximum angle of attack of 15◦ .
3.2
Axisymmetric flows
In all axisymmetric test cases 200 equidistant points were considered on body surface for shock-expansion
tool, where as 200 X 75 X 32 cells grids were used over half-body for inviscid CFD analysis.
3.2.1 Cones with different semi-vertex angles
Right circular cones with semi-vertex angles of 10◦ , 15◦ and 20◦ were used to test at a free-stream Mach
number of 2.72. Figure 6(a) shows the Euler grid used for inviscid CFD analysis and figure 6(b) shows
pressure contours with streamlines on right circular cone with semi-vertex angle of 20◦ . The experimental results of Yaholam [11] were used for comparison purpose, which were conducted at Reynolds
number of 5.5X105 . Table 1 shows the comparison of Cp between shock-expansion, experimental and
Euler CFD results for different right circular cones considered. Shock-expansion results underpredicted,
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
-0.4
Angle of attack = 0
-0.8
o
Airfoil upper surface
Angle of attack = 5
o
-0.2
-0.6
0
(P - P ∞)/P ∞
(P - P ∞)/P ∞
-0.4
0.2
0.4
-0.2
0.6
0
0.8
1
McLellan et. al.
Shock - expansion
Euler CFD
0
-1
20
40
60
80
100
McLellan et. al.
Shock - expansion
Euler CFD
0.2
0
20
40
60
80
100
Location as percentage of chord
Location as percentage of chord
(a) Pressure distribution at aoa = 0◦
(b) Upper surface pressure distribution at aoa = 5◦
-1
Airfoil upper surface
Angle of attack = 10
o
-0.8
(P - P ∞)/P ∞
(P - P ∞)/P ∞
-0.8
-0.6
-0.6
-0.4
Airfoil upper surface
-0.4
Angle of attack = 15
McLellan et. al.
Shock - expansion
Euler CFD
-0.2
0
20
40
60
80
-0.2
100
o
McLellan et. al.
Shock - expansion
Euler CFD
0
20
40
60
80
100
Location as percentage of chord
Location as percentage of chord
(c) Upper surface pressure distribution at aoa = 10◦
(d) Upper surface pressure distribution at aoa = 15◦
-1.1
-1
Airfoil lower surface
0
-1
1
2
-0.9
3
4
-0.8
(P - P ∞)/P ∞
(P - P ∞)/P ∞
5
-0.7
-0.6
6
7
8
9
McLellan et. al. Aoa = 5 o
McLellan et. al. Aoa = 10 o
o
McLellan et. al. Aoa = 15
McLellan et. al. Aoa = 20 o
Shock - expansion Aoa = 5 o
o
Shock - expansion Aoa = 10
Shock - expansion Aoa = 15 o
Shock - expansion Aoa = 20 o
o
Euler CFD Aoa = 5
Euler CFD Aoa = 10 o
Euler CFD Aoa = 15 o
o
Euler CFD Aoa = 20
10
-0.5
11
Airfoil upper surface
-0.4
Angle of attack = 20
12
o
13
McLellan et. al.
Shock - expansion
Euler CFD
-0.3
-0.2
0
20
40
14
15
60
80
100
16
0
20
40
60
80
Location as percentage of chord
Location as percentage of chord
(e) Upper surface pressure distribution at aoa = 20◦
(f) Lower surface pressure distribution
100
Figure 2: Pressure distribution on 5% thick diamond airfoil at free-stream Mach number of 6.86 and
different angle of attacks
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
(PP/P_inf
- P ∞)/ P ∞
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
(b) Pressure contours with streamlines at aoa = 20◦
(a) Euler grid around airfoil
Figure 3: Euler grid, pressure contours on 10% thick bi-convex airfoil at free-stream Mach number of
6.86 and 20◦ angle of attack
the experimental results with in 4% error and Euler CFD results with in 5.4% error. Euler CFD results
slightly overpredicted, when compared to experimental ones.
Table 1: Comparison of Cp for different cone semi-vertex angles at M∞ = 2.72 and Re = 5.5X105
Cone semi-vertex
angle (Deg.)
10
15
20
Experimental
Shock-expansion
Euler CFD
0.09526
0.18091
0.29523
0.09089
0.17862
0.29169
0.09615
0.18565
0.30055
3.2.2 Tangent-ogive cylinder with fineness ration of 3
Tangent-ogive cylinder with fineness ration of 3 and ogive radius of 9.25 times the diameter of cylinder
is considered to test the tool at free-stream Mach numbers of 3, 4.25, 5.05 and 6.3. Figure 7(a) shows the
Euler grid used for inviscid CFD analysis and figure 7(b) shows pressure contours with streamlines on
tangent-ogive at free-stream Mach number of 6.3. Experimental results of Eggers et al [12] were used
for comparison, which were conducted at NASA Ames supersonic wind tunnel at Reynolds numbers of
1.09X106 at Mach numbers 3 and 4.25, 0.52X106 at Mach number 5.05 and 0.22X106 at Mach number
6.3. Figure 7(c) to (f) shows the comparison of Cp between shock-expansion, experimental and Euler
CFD results at the free-stream Mach numbers considered. There is good agreement between experimental and Euler CFD results at almost all Mach numbers, where as shock-expansion results underpredicted
at all Mach numbers. The deviation decreased with increasing Mach number.
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
-1
Angle of attack = 0
-1
o
-0.5
-0.8
-0.4
1
-0.2
(P - P ∞)/P ∞
-0.6
(P - P ∞)/P ∞
0
0.5
1.5
Airfoil upper surface
Angle of attack = 6
o
0
0.2
2
0.4
2.5
0.6
3
0.8
3.5
1
4
McLellan et. al.
Shock - expansion
Euler CFD
4.5
5
20
-1
40
60
McLellan et. al.
Shock - expansion
Euler CFD
1.2
1.4
80
0
20
40
60
80
100
Location as percentage of chord
Location as percentage of chord
(a) Pressure distribution at aoa = 0◦
(b) Upper surface pressure distribution at aoa = 6◦
-1
Airfoil upper surface
Angle of attack = 10
o
-0.8
-0.8
-0.6
(P - P ∞)/P ∞
(P - P ∞)/P ∞
-0.4
-0.6
-0.2
0
-0.4
Airfoil upper surface
Angle of attack = 15
o
0.2
McLellan et. al.
Shock - expansion
Euler CFD
0.4
0
20
40
60
80
McLellan et. al.
Shock - expansion
Euler CFD
-0.2
100
0
20
40
60
80
100
Location as percentage of chord
Location as percentage of chord
(c) Upper surface pressure distribution at aoa = 10◦
(d) Upper surface pressure distribution at aoa = 15◦
-1.1
Airfoil lower surface
0
2
-1
4
6
-0.9
(P - P ∞)/P ∞
(P - P ∞)/P ∞
8
-0.8
-0.7
-0.6
-0.5
-0.4
20
40
60
12
16
Angle of attack = 20 o
18
80
100
o
McLellan et. al. Aoa = 6
o
McLellan et. al. Aoa = 10
o
McLellan et. al. Aoa = 15
o
McLellan et. al. Aoa = 20
o
Shock - expansion Aoa = 6
o
Shock - expansion Aoa = 10
o
Shock - expansion Aoa = 15
o
Shock - expansion Aoa = 20
o
Euler CFD Aoa = 6
o
Euler CFD Aoa = 10
o
Euler CFD Aoa = 15
o
Euler CFD Aoa = 20
14
Airfoil upper surface
McLellan et. al.
Shock - expansion
Euler CFD
0
10
20
22
24
0
20
40
60
80
Location as percentage of chord
Location as percentage of chord
(e) Upper surface pressure distribution at aoa = 20◦
(f) Lower surface pressure distribution
100
Figure 4: Pressure distribution on 10% thick bi-convex airfoil at free-stream Mach number of 6.86 and
different angle of attacks
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
P / P∞
P/P_inf
11
10
9
8
7
6
5
4
3
2
1
(a) Euler grid around all-body hypersonic aircraft
2.2
(b) Pressure contours with streamlines at aoa = 15◦
13
All-body hypersonic aircraft pitch plane leeward side
All-body hypersonic aircraft pitch plane winward side
12
2
o
o
Lockman et. al. Aoa 0
Lockman et. al. Aoa 5 o
o
Lockman et. al. Aoa 10
o
Lockman et. al. Aoa 15
Shock - exp. Aoa 0 o
o
Shock - exp. Aoa 5
o
Shock - exp. Aoa 10
o
Shock - exp. Aoa 15
Euler CFD Aoa 0 o
o
Euler CFD Aoa 5
o
Euler CFD Aoa 10
o
Euler CFD Aoa 15
1.8
1.6
1.4
10
9
8
P/P ∞
P/P ∞
1.2
1
0.8
7
6
5
0.6
4
0.4
3
0.2
2
0
1
-0.2
Lockman et. al. Aoa 0
Lockman et. al. Aoa 5 o
o
Lockman et. al. Aoa 10
o
Lockman et. al. Aoa 15
Shock - exp. Aoa 0 o
o
Shock - exp. Aoa 5
o
Shock - exp. Aoa 10
o
Shock - exp. Aoa 15
Euler CFD Aoa 0 o
o
Euler CFD Aoa 5
o
Euler CFD Aoa 10
o
Euler CFD Aoa 15
11
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
Location, x/l
Location, x/l
(c) Pressure distribution on upper surface
(d) Pressure distribution on lower surface
1
Figure 5: Euler grid, pressure contours and pressure distribution on all-body hypersonic aircraft at freestream Mach number of 7.4 and different angle of attacks
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
CP
P/P_inf
0.28
0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
(a) Euler grid on cone
(b) Pressure contours with streamlines on 20◦ cone
Figure 6: Euler grid, pressure contours on right circular cone with semi-vertex angle of 20◦ at a freestream Mach number of 2.72
3.2.3 Cone-cylinder-flare configuration
Cone-cylinder-flare configuration is considered to test the tool at free-stream Mach numbers of 1.75, 2, 3,
4 and 4.5. Figure 8(a) shows the Euler grid used for inviscid CFD analysis and figure 8(b) shows pressure
contours with streamlines on the configuration at free-stream Mach number of 4.5. Experimental results
of Washington et al [13] were used for comparison, which were conducted at supersonic wind tunnel at
Aberdeen, Maryland at Reynolds number of about 0.45X106 . Figure 8(c) to (f) shows the comparison
of Cp between shock-expansion, experimental and Euler CFD results at the free-stream Mach numbers
considered. There is good agreement between experimental and Euler CFD results at almost all Mach
numbers. The shock-expansion results deviated more from CFD and experimental results at all Mach
numbers on cylinder and flare sections, which can be attributed to neglected interaction of outwardrunning Mach waves with inward-running Mach waves from the shock and stream lines. This interaction
reduces with increasing Mach number and hence the deviation decreased with increasing Mach number.
4 Conclusion
In the present study Shock-Expansion tool was developed and applied to three 2-D bodies at different
angle of attacks and three axisymmetric bodies at zero angle of incidence for different Mach numbers.
In all the cases, the results obtained are very much in good agreement with Euler CFD results, except
for cone-cylinder-flare configuration, which can be attributed to influence of reflection Mach waves. In
all the cases, the trends of experimental results were predicted. The time taken by shock-expansion
calculations is very negligible, when compared with Euler CFD simulations. Moreover shock-expansion
calculation doesn’t need any grid, which saves grid generation time too. Hence quick inviscid estimates
can be generated with reasonable accuracy. The deviation between the results of 2-D bodies and their
corresponding experimental results increased with increase of angle of attack. These deviations can be
attributed to increasing viscous effects with angle of attack on suction side, like separation. There is
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
better agreement on pressure side even at high angle of attacks. For axisymmetric case the accuracy of
prediction increased with free-stream mach number.
Acknowledgments
This work has been supported by the Council of Scientific and Industrial Research, India. Thanks are due
to Mr. Vajjala Keshava Suman, Scientist, CTFD Division, CSIR-NAL for helpful discussions throughout
this work.
References
[1] Anderson, John David, Hypersonic and high temperature gas dynamics, McGraw-Hill series in
Aeronautical and Aerospace Engineering, 1989.
[2] Maurice Rasmussen, Hypersonic flow, John Wiley & Sons, Inc., New York, 1994.
[3] Epstein, S. Paul, On the Air Resistance of Projectiles, Proc Natl. Acad. Sci. USA, 17(9), 532547,
Sep 1931.
[4] Eggers, A. J., Syvertson, C. A., and Kraus, S., A Study of Inviscid Flow About Airfoils at High
Supersonic Speeds, NACA Report 1123,1953.
[5] Syvertson, C. A., and Dennis, D. H., A Second-Order Shock Expansion Method Applicable t Bodies of Revolution Near Zero Lift, NACA TR 1323,1957.
[6] Jackson, C. M., Sawyer, W C., and Smith, R. S., A Method for Determining Surface Pressures on
Blunt Bodies of Revolution at Small Angles of Attack in Supersonic Flow, NASA TN P-4865,1968.
[7] Dejarnette, F. K., and Ford, C. P., A New Method for Calculating Surface Pressures-on Bodies at an
Angle of Attack in Supersonic Flow, Proceedings of the 11th Navy’ Symposium on Aerobaffistics,
Vol. 11, August 22-24, 1978, pp. 199-231.
[8] Moore, Frank G., Armistead, Michael A., Rowles, Steve H., and Dejarnette Fred K., Second-order
Shock-Expansion Theory Extended to Include Real Gas Effects, NAVSWCTR 90-683, 1992.
[9] McLellan, Charles H., Bertram, Mitchel H., and Moore, John A., An Investigation of Four Wings
of Square Plan Form at a Mach number of 6.86 in the Langley 11-Inch Hypersonic Tunnel, NACA
RM L51D17,1951.
[10] Lockman, William K., Lawrence, Scott L., Cleary, Joseph W., Experimental and Computational
Surface Flow-Field Results for an All-Body Hypersonic Aircraft, AIAA paper No. 90-3067, 1990.
[11] Yahalom Rafael, An Experimental Investigation of Supersonic Flow Past Yawed cones, Report No.
AS-71-2, College of Engineering, University of California, Berkeley, 1971.
[12] Eggers, A. J. Jr., and Savin, Raymond C., A Unified Two-Dimensional Approach to the Calculation
of Three-Dimensional Hypersonic Flows, With Application to Bodies of Revolution, NACA Report
1249, 1952.
[13] Washington, William D., and Humphery, James A., Pressure Measurements on Four ConeCylinder-Flare Configurations at Supersonic Speeds, AD 699359, 1969.
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
CP
P/P_inf
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
(a) Euler grid on tangent-ogive
0.3
(b) Pressure contours with streamlines at Mach = 6.3
0.3
M∞ = 3
M ∞ = 4.25
l/d = 3
l/d = 3
K = 1.00
0.25
0.25
0.2
K = 1.42
Eggers at. al.
Shock - expansion
Euler CFD
Pressure coefficient, Cp
Pressure coefficient, Cp
Eggers et. al.
Shock - expansion
Euler CFD
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x/l
(c) Cp distribution at Mach = 3
0.3
0.6
0.3
M ∞ = 5.05
M ∞ = 6.30
l/d = 3
0.25
K = 1.68
K = 2.10
Eggers at. al.
Shock - expansion
Euler CFD
Pressure coefficient, Cp
Pressure coefficient, Cp
Eggers at. al.
Shock - expansion
Euler CFD
0.2
1
(d) Cp distribution at Mach = 4.25
l/d = 3
0.25
0.8
x/l
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
0
0.2
0.4
0.6
0.8
x/l
(e) Cp distribution at Mach = 5.05
1
0
0.2
0.4
0.6
0.8
x/l
(f) Cp distribution at Mach = 6.3
Figure 7: Euler grid, pressure contours and Cp distribution on tangent-ogive cylinder with fineness ratio
of 3 at different free-stream Mach numbers
16th Annual CF D Symposium, August 11 − 12, 2014, Bangalore
P/ P ∞
P/P_inf
1.45
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8
(a) Euler grid on cone-cylinder-flare
1.2
M ∞ = 1.75
5
Re = 4.61 X 10
(b) Pressure contours with streamlines at Mach = 4.5
1.3
Washington et. al.
Shock-expansion
Euler CFD
M∞ = 3
5
Re = 4.56 X 10
Washington et. al.
Shock-expansion
Euler CFD
1.25
1.15
1.2
1.1
1.15
1.1
1.05
P/P ∞
P/P ∞
1.05
1
1
0.95
0.95
0.9
0.9
0.85
0.8
0.85
0.75
0.8
0
1.4
5
10
15
0.7
0
5
10
Distance from nose tip in inches
Distance from nose tip in inches
(c) Pressure distribution at Mach = 1.75
(d) Pressure distribution at Mach = 3
M∞ = 4
5
Re = 4.33 X 10
1.5
Washington et. al.
Shock-expansion
Euler CFD
15
M ∞ = 4.5
5
Re = 4.17 X 10
1.4
1.3
Washington et. al.
Shock-expansion
Euler CFD
1.3
1.2
1.2
P/P ∞
P/P ∞
1.1
1.1
1
1
0.9
0.9
0.8
0.7
0.8
0
5
10
15
0.7
0
5
10
Distance from nose tip in inches
Distance from nose tip in inches
(e) Pressure distribution at Mach = 4
(f) Pressure distribution at Mach = 4.5
15
Figure 8: Euler grid, pressure contours and pressure distribution on cone-cylinder-flare configuration at
different free-stream Mach numbers