Investigation of a Single-Jack Flexible Supersonic Nozzle

K. A. Juhany * and H. E. Husaini,†

King Abdulaziz University, Jeddah, 21589, Saudi Arabia

An alteration is presented that would allow a single-jack flexible nozzle achieve a higher Mach number range within a confined length. CFD analysis is used in the investigation to provide the flow uniformity exiting the nozzle. A Mach number range of 1.5<M<2.5 was increased to 1.5 <M< 3.5 by merely drawing the nozzle walls closer together to achieve a higher area ratio and thus a higher Mach number.

Nomenclature M = Mach Number θ1 = Slope Angle At = Test section area Ao = Throat area R = The radius at the throat A1 = Area at the inflection point λ = Strait line length ỳo = Half throat height of original nozzle yo = Half throat height β = Beam taper ratio ζ = Throat area R = The radius at the throat A1 = Nondimensional length ho = Minimum thickness of the flexible plate B = As defined in eq.2 C = As defined in eq.3

I. Introduction INGEL-JACK variable Mach number nozzles1 remain the simplest type of its class and continue to present the most economical means of providing a wide range of supersonic Mach numbers. The SVM150 is the original

wind tunnel built in 1955 to demonstrate the single-jack concept. It is planned to install the tunnel in the student aerodynamic laboratory at King Abdulaziz University. The SVM 150 is a blow-down type tunnel with a rectangular test section of 150 x 150 mm. The SVM150 employs a single-jack variable Mach number nozzle with a Mach range of 1.5≤ M ≤2.5. It is proposed to extend the Mach number range of the tunnel beyond the current limit using the same type of nozzle. Such changes must take place while maintaining the tunnel components unchanged. However, internal parts of the nozzle section must be modified to provide the desired outcome. The Mach number range can be extended using the same technique provided in Ref. 1. Nevertheless, this would entail an increase in nozzle length beyond the existing space. Several concepts are proposed to extend the Mach number range using the same nozzle section and length of the SVM150, while maintaining the simplicity of the single-jack

S

Figure 1. The single-jack variable Mach number nozzle1.

* Assistant Professor, Aeronautical Engineering Dept., E-mail:khalid@alumni.caltech.edu, life time member. † Student, Aeronautical Engineering Dept.

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45th AIAA Aerospace Sciences Meeting and Exhibit8 - 11 January 2007, Reno, Nevada

AIAA 2007-958

Copyright © 2007 by K. A. Juhany. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

design. The Foelsh2 method is applied to obtain the Mach number contours while the CFD commercial code FLUENT is employed to validate the uniformity of the flow exit.

II. Single-Jack Nozzle Design

A. Single-Jack Concept The single-jack nozzle design, shown in Fig. 1, consists of contoured throat blocks mounted on pivot arms.

Flexible plates are attached to the throat blocks and fixed at the test section plates. The flexible plates form the variable contour of the nozzle walls. The change in the load located at the throat causes symmetric rotation of the pivot arms about the node. As a consequence, the throat height reduces and the flexible plate deforms to form the desired nozzle contour. Depending on the location of the node with respect to the throat blocks, the nozzle plate length, and the choices of the plate thickness profile, a parallel flow can be obtained at desired Mach number. Further description of the single-jack nozzle is included in the appendix.

B. Basic Design Consideration The nozzle length of the current design is required to have the

same nozzle length of the SVM150 tunnel. Further, it is required to extend the Mach number range to Mach 3.5. This requirement further complicates the design since higher Mach number nozzles require greater nozzle length. An initial approach is to introduce a change in the nozzle inlet. To partially submerge the subsonic portion of the nozzle in the stagnation chamber would allow a longer nozzle to be installed. Figure 2 demonstrates the difference between the original SVM 150 inlet design and the proposed design. The leading edge of the nozzle inlet must be carefully designed sot that separation at the leading edge doesn’t take place. Changing the nozzle inlet design would move the throat nearer to the inlet of nozzle housing and thus the nozzle length can be increase by at least 65mm. This additional 65 mm will allow a new nozzle with a terminal section of 380 mm (in length).

Figure 2. a) Original SVM150 nozzle inlet design, b)Proposed nozzle inlet design.

300

320

340

360

380

400

420

440

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7

M

L (m

m)

θ = 6 deg θ = 8 deg θ = 10deg

Figure 3. Length of terminal section at different Mach number and slop angle.

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To start designing the nozzle the slope angle of the flexible wall must be designated. Also, an intermediate Mach number within the desired range required (l.5<M<3.5). An intermediate Mach number of 3.0 is selected. To define the slope angle, a parametric study is required. As the slope angle decreases the nozzle length increases and the opposite is true. However, since the nozzle has to be as short as possible the slope angle must be made as large as possible. But how large can it be? A slope angle of above 10 degrees or even slightly less would cause a large deviation between the Foelsch profile and beam profile. This deviation would have an adverse effect on the uniformity of the

flow at the exit of the nozzle. Following the Foelsch analysis a plot can be produced between slope angle and Mach number and the length of terminal section. This plot is shown in Fig.3 for different slope angles.

Three parameters are required to determine the nozzle shape; the Mach number, the slope angel, and the nozzle length. With the aid of Fig. 3 the intermediate Mach number and the desired length of 380 mm would result in a slope angel of 8°. The selected parameters are entered in the Foelsh method to obtain the Mach number contours and in beam theory to obtain the beam profile. The outcome is displayed in Fig. 4 which shows both profiles and the deviation between them. Once the beam profile is determined it is used as the terminal section of the new nozzle. Therefore, the entire nozzle contour is defined and shown in (Fig. 5).

To validate the uniformity of the exit flow and the Mach number distribution of the nozzle a CFD analysis is executed using the commercial code Fluent. Pressure-inlet boundary condition was applied at the inlet of the nozzle (both total pressure and static pressure at the inlet are specified) while pressure-outlet was chosen for the exit (the exit pressure is specified). 2-D coupled implicit steady solver with inviscid model was used. The grid used was a structured grid. The problem is solved initially with a coarse grid, and then the adaptation of the grid is executed until a grid independent solution is

obtained.

40

45

50

55

60

65

70

75

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

Y (m

m)

Beam profile Foelsch profile

Figure 4. Deviation between foelsch and beam profile

0 200 400 600 800

020406080

Figure 5. Nozzle contour from throat to test

2.95

2.96

2.97

2.98

2.99

3

-0.075 -0.045 -0.015 0.015 0.045 0.075

Position (m)

Mac

h N

umbe

r

-0.02

-0.01

0

0.01

0.02

-0.075 -0.045 -0.015 0.015 0.045 0.075

Position (m)

Velo

city

Ang

le (d

eg)

(a) (b)

(c) (d)

Figure 6. a) The Mach number distribution along the nozzle at design Mach number M = 3. b)The type of grid used. c) The Mach number distribution at the test section. d) The Mach Angle at the test section.

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The maximum total number of cells used the grid is 445200. The grid used, the Mach number contour along with the uniformity of the exit flow is shown in Fig. 6.

The Mach number is varied by changing the single-jack load. As the jack load is varied the nozzle contour varies. Fig 7 shows selected nozzle contours whose load magnitude and slope angle is specified in Table 1. Table 2 shows a summary of the computational results of the nozzle contours specified in Fig 7. The table shows the exit Mach number and the maximum deviation of the flow from the axial direction. An acceptable level3 is also indicated in the table for direct comparison. The Mach number distribution at the nozzle exit is shown in Fig. 8 for the extremes of the nozzle Mach number range (1.5 and 3.5).

It is concluded that the nozzle with the selected parameters does not produce the required quality, since the value of ΔM/M (± 2.5 %) exceeds the acceptable value (± 0.8 %). Several parameters were changed to attempt to improve the Mach number distribution at the nozzle exit. The aforementioned analysis follows that of Ref.1, it was concluded that in order to achieve the desired flow quality, the nozzle length has to be extended beyond the available space or a different approach has to be undertaken. A different approach is proposed that would still use the simplification of a single-jack design with a combined nozzle displacement in the direction normal to the flow axis.

Load P (kN) Mach number Slope angle (degree)

0.572 1.5 2.085 1.152 2.5 4.25 2.217 3 8 2.829 3.5 10.2

Table.1 Mach number at test section and slop angle at different single load.

Mach

number

(ΔM /

M)

(ΔM / M) acceptabl

e

velocity

angle at the test section

velocity angle

acceptable

1.5

± 0.33 %

± 0.4 %

0 deg

± 0.1 deg 3

± 0.45 %

± 0.6 %

0.001 deg

± 0.1 deg

3.5

± 2.5 %

± 0.8 %

0 deg

± 0.1 deg

Table.2 Evaluations of all contours at different Mach number depend on the results come from CFD.

3.34

3.39

3.44

3.49

3.54

-0.075 -0.045 -0.015 0.015 0.045 0.075

Position (m)

Mac

h N

umbe

r

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1X

Y(m

m)

M=3.5 M=3 M=2.5 M=1.5

Figure 7. different beam profiles at different Mach number at test section.

(a)

1.49

1.495

1.5

1.505

-0.075 -0.045 -0.015 0.015 0.045 0.075

Position (m)

Mac

h N

umbe

r

(b) Figure 8. The Mach number distribution at the test section. a) at Mach 3.5 contour b) at Mach 1.5 contour.

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C. Nozzle Displacement Method The following methodology was applied to conceptualize the improved nozzle over the one discussed in previous section. According to the Foelsch method, any nozzle contour must satisfy the following expression:

R 1 cos θ1( )−( )⋅ λ s+( ) sin θ1( )⋅+

R 1 cos θ1( )−( )⋅ λ sin θ1( )⋅+

AtAo

1−

sin θ1( )θ1

A1Ao⋅ 1−

(1)

The expressions on the left hand side and right hand side of the equation will be referred to as B and C, respectively. Thus:

BR 1 cos θ1( )−( )⋅ λ s+( ) sin θ1( )⋅+

R 1 cos θ1( )−( )⋅ λ sin θ1( )⋅+ (2)

and

C

A tA o

1−

sin θ 1( )θ 1

A 1A o

⋅ 1−

(3)

For true Foelsch profiles the values of B and C have to be the same. Figure 9 plots both values of B and C at different slope angle with the Mach number as a parameter.

This chart indicates that at the values of B (which is independent of the Mach number) of about 1.68 the corresponding C values are nearly the same for the Mach number range of 2.5 to 3.5, while the slope angle is nearly the same (8 degrees). Since the B and C are the same and the slope angle is also nearly the same for Mach numbers 2.5, 3.0 and 3.5 then they must have the same Foelsh profile, i.e. they have nearly identical wall profiles. The only way to change the Mach number for identical contours is to change the area ratio. Therefore, by using the same shape of nozzle with same slope angle we can get different Mach Numbers (ranging from 2.5 to 3.5) at test section by simply shifting the nozzle walls closer together or farther apart (up and down motion).

To validate the above concept the nozzle contours were studied computationally using the same technique discussed above. Starting with the Mach 3.0 nozzle contour, the contour was shifted in two opposite vertical directions to vary the area ratio. Table3 and (Fig. 10) show the results of CFD program. The values of the nozzle exit Mach number and angular deviation fall within the acceptable level. Therefore, a nozzle that combines both the single-jack concept and the ability of the nozzle to displace vertically would produce the desired flow quality. Mach numbers below 3.0 or 2.5 can be varied using a single-jack, while Mach numbers greater than 2.5 to 3.5 can be obtained by simply displacing the nozzle in the vertical direction while retaining the same contour of Mach 2.5 or 3.0. Such a nozzle design is in progress and the results of will be presented in a future conference. It should be added that the reduction of the nozzle height would not require a reduction in model size (model length). This due to the fact that the test rhombus increased as the Mach number is increased.

11.11.21.31.41.51.61.71.81.9

2

2 3 4 5 6 7 8 9 10slop angle (degree)

B a

nd C

B C at M =1.5 C at M=2 C at M=2.5 C at M =3

C at M=3.5 C at M=4 C at M=4.5 C at M=5

Figure 9. Intersections between B and C at different mach number and slop angle.

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III. Conclusion An increase in Mach number range of an existing

wind tunnel (SVM150) utilizing a single-jack nozzle is investigated. The original Mach number range of 1.5 < M < 2.5 is to be extended to 1.5 < M < 3.5. The limitation is the existing short length of the test section. CFD code Fluent was used to aid in the nozzle deign and by determining the flow quality of the resulting nozzle contours. Modifying the subsonic inlet of the nozzle did provide some additional length. However, the Mach number variation across the nozzle exit was 2.5% which is higher than the maximum desired value of 0.8%. A parametric study of the Foelsh profiles indicated that for a fixed nozzle length, the nozzle terminal section of the Mach number range of 2.5<M<3.5 are approximately the same slope angle. Thus, study was made by fixing the

nozzle contour at Mach 3 and displacing the nozzle walls in the normal direction to the flow. The CFD study indicates that desired Mach number distribution is attained by this method.

Appendix

A. Description of The Single-Jack Nozzle Figure A1 shows that the nozzle consists of a subsonic segment IO and a supersonic segment OFAQ. The

segment IOF is an arbitrary curve that must satisfy certain geometric boundary conditions. The curve IOF is usually taken as circular with a radius R of the range 10<R/Y0< 20, where Y0 is the half the throat height located at O. The segment OF starting at the throat is a circular arc connected to a straight segment FA that bounds a supersonic region of radial streamlines. The segment FA has slope angle θ1 and length λ. The nozzle design Mach number and the arbitrary choice of the slope angle θ1 determine the location of point A. The segment IOFA is physically a solid block, the terminal curve or Foelsch profile AQ is a flexible part which is considered as the neutral line of a beam with a linear thickness distribution as shown in Fig. A2. An important property of the elastic beam is that the slope of the elastic line at point A will intersect at one point called the node, defend as the center of convergence of the tangent from point A when this is deflected by a point load at A.

3.44

3.45

3.46

3.47

3.48

3.49

-0.075 -0.045 -0.015 0.015 0.045 0.075

Position (m)

Mac

h N

umbe

r

(a)

2.47

2.48

2.49

2.5

2.51

-0.075 -0.045 -0.015 0.015 0.045 0.075

Position (m)

Mac

h N

umbe

r

(b) Figure 10. The Mach number distribution at the test section. a) at Mach 3.5 contour, b ) at Mach 2.5 contour.

Mach number

(ΔM /

M)

(ΔM / M) acceptable

velocity angle at the test section

velocity angle

acceptable

2.5

0.44 %

± 0.55 %

± 0.005 deg ± 0.1 deg

3

± 0.45 %

± 0.6 %

0.001 deg ± 0.1 deg

3.5

0.72 %

± 0.8 %

0.002 deg ± 0.1 deg

Table.3 Evaluation of all contours at different Mach number depend on the results come from CFD.

Figure A1 . Design feature of supersonic nozzle profile.

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If a point load is applied at point O and the beam is pivoted at the node of the and its end is fixed at point Q (located at the inlet of the test section) a nozzle contour is formed. A successful design is achieved when the elastic

line of the beam matches the Foelsch profile.

Figure A2 . Flexible beam deflection.

Acknowledgments We would like to express our sincere gratitude to Mr. John Rosen, the inventor of the single-jack variable Mach

number nozzle, for his encouragement, support, and keen interest in our work that lead to his generous donation of the critical sections of the SVM150 tunnel.

References 1Rosén, J., “The Design and Calibration of a Variable Mach Number Nozzle,” Journal of Aeronautical Society, Vol.22, No.

7, 1955, pp. 484-490. 2Foelsch, K., “A New Method of designing two-dimensional Laval nozzle,” North American aviation, Inc., Report NA-

46235-2. March 1946. 3Pope, A. and Goin, K. L., High Speed Wind Tunnel Testing, Robert E. Krieger publishing co, New Mexico, 1965.

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