[american institute of aeronautics and astronautics 26th aiaa aerodynamic measurement technology and...

American Institute of Aeronautics and Astronautics

1Approved for public release; distribution is unlimited.

On the Importance of Contraction Design for Supersonic Wind Tunnel Nozzles*

Frederick L. Shope* and Moufid E. Aboulmouna† Aerospace Testing Alliance

Arnold Engineering Development Center, Tennessee

CFD solutions were computed to decide whether the contraction contour has significant effect on the flow uniformity in the test section of an axisymmetric supersonic wind tunnel nozzle. The supersonic portion of a nozzle contour is usually designed with an aerodynamically rigorous method of characteristics, whereas the contraction contour is typically a convenient smooth mathematical function that has not been rigorously derived from aerodynamics theory. The design procedure for the supersonic contour must assume some flow profile in the throat as a boundary condition, but whether standard contraction contours deliver the assumed flow profile is not documented. A set of rules for good contraction design seems to have accumulated over the years, but it is unclear whether the rules are either necessary or sufficient. It is known from method of characteristics (MOC) analysis that the throat flow profile has an observable effect on flow quality. A question then is whether it would be useful to develop an aerodynamically rigorous contraction design procedure. In the present analysis, inviscid flow solutions were computed for axisymmetric nozzles with various contraction contours that selectively violate rules on maximum inflection angle and avoidance of any curvature discontinuities. The inviscid choice was made specifically for consistency with the standard inviscid MOC design procedure. Solutions were computed for the Arnold Engineering Development Center (AEDC) Tunnel C Mach 4 nozzle contour minus the boundary-layer correction. The contractions examined were (1) the original cylinder-quartic-cone-quartic as developed by James C. Sivells at AEDC, (2) a model based on sinnx, and (3) a double ellipse model. Large excursions in inflection angle and curvature discontinuity were examined. Flow profiles were computed at the nozzle throat and at the nozzle exit. Results were compared to a baseline contraction, specifically the Tunnel C contraction contour as originally designed by Sivells. The primary finding was that large inflection angles and curvature discontinuities upstream of the throat do not significantly degrade test section flow uniformity if the curvature at the throat is continuous. Sivells' contraction model does deliver the throat flow profile assumed by his MOC design code. The results to date do not support the need to develop an aerodynamically rigorous contraction contour design procedure. However, the issue should be revisited if a nozzle design procedure is developed that relaxes some of the constraints in the current MOC procedure.

I. Introduction HE design of an axisymmetric supersonic wind tunnel nozzle contraction, that is, the subsonic converging duct upstream of the throat, has traditionally been accomplished with methods that are aerodynamically much less

* The research reported herein was performed by the Arnold Engineering Development Center (AEDC), Air Force Materiel Command. Work and analysis for this research were performed by personnel of Aerospace Testing Alliance (ATA), the operations, maintenance, information management, and support contractor for AEDC. Further reproduction is authorized to meet the needs of the U.S. Government. * Engineer, Integrated Test and Evaluation Dept., 676 Second St., Arnold AFB TN 37389-4001, Senior member AIAA. † Engineer, Integrated Test and Evaluation Dept., 1099 Schriever Ave., Arnold AFB TN 37389-9013.

T

26th AIAA Aerodynamic Measurement Technology and Ground Testing Conference<BR>23 - 26 June 2008, Seattle, Washington

AIAA 2008-3940

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.



rigorous than techniques for the supersonic contour. At the Arnold Engineering Development Center (AEDC), the contraction contour is typically a geometrically convenient function designed to have certain smoothness characteristics that are said to promote good flow uniformity at the nozzle test section. One such function is based on a sequence of quartic polynomials and cones that provides continuous curvature all along the contraction contour. The usual procedure for designing a supersonic nozzle is to design the supersonic portion of the contour first, typically using a rigorous method of characteristics (MOC), and then designing the contraction geometrically to be compatible and smooth in joining the supersonic contour at the throat. The supersonic design is based on an inviscid space-marching MOC and provides a direct computation of the contour from a single sweep of the flow field. No such convenient method is available for the subsonic portion of the contour primarily because the flow equations are elliptic (rather than hyperbolic, as in supersonic flow) and are not amenable to a convenient space-marching procedure. Accordingly, contraction design is usually based on aesthetically pleasing mathematical functions that are tested against empirical criteria, e.g., for flow separation. To the authors' knowledge, there is no documented evidence that the subjective rules that have developed around contraction design are either necessary or sufficient for good flow quality. The AEDC-standard contraction contour based on a cylinder-quartic-cone-quartic (CQCQ) appears to be a valid approach, but it is very limited in the range of geometries it can accommodate. The purpose of the present work is to scrutinize with computations the various rules and assumptions associated with contraction design practice to determine whether they are valid or necessary.

A. Motivation As motivation for the present work, MOC solutions were computed for the AEDC Tunnel C Mach 4 nozzle,

sometimes referred to as the "aerothermal tunnel" (Fig. 1). This nozzle was designed by J. C. Sivells using his MOC design code, which is documented in Ref. 1. Sivells designed the contraction based on his CQCQ model. The motivational solutions were computed to illustrate the effect of different assumptions for the starting flow profile at the throat. The solutions were computed using an irrotational, space-marching method of characteristics. The equation of state is for a thermally and calorically perfect gas. The code, designated NOZMOCA, has not been documented in the open literature. The solutions were computed for the inviscid contour (i.e., before the boundary-layer correction was added). Two solutions were computed, each with a different assumption on the throat flow starting solution. The first solution used the Hall-Kliegel-Levine (HKL) analytical series solution for irrotational flow in a circular arc throat (as given in Ref. 1). The sonic line and throat characteristic from the HKL solution are shown in Fig. 2. For the present MOC solution, the HKL model was evaluated along a vertical line 1 in. downstream of the throat. The HKL model is also that used by Sivells in his design code to construct a throat characteristic. The second MOC solution used a radially uniform source flow over a spherical surface at a specified location near the throat. Both start lines are plotted in Fig. 3. Some results from these two solutions are shown in Figs. 4 through 9. Figure 4 shows the characteristic net for the HKL-initialized solution. Note the uniformity of the characteristic spacing. A blowup of the throat region is shown in Fig. 5. The equivalent results for the source-flow-initiated solution are given in Figs. 6 and 7. Note that the characteristic net for the source-flow solution (Fig. 6) shows dramatic irregularities in spacing. The origin of these irregularities is visible in the throat blowup of Fig. 7. These irregularities manifest themselves in the centerline static pressure distributions shown in Fig. 8. Clearly, the source-flow distribution shows undesirable effects near the throat and entering the test rhombus. Figure 9 compares the Mach number profiles at the nozzle exit for the two cases. While both cases have generally good Mach number uniformity, the HKL solution is dramatically better than the source-flow solution.

The conclusion is that the throat flow profile has a significant effect on the downstream flow. It should be observed that the source-flow profile is not necessarily unphysical, but merely a possible flow different from the HKL profile. In the authors' (unproven) opinion, the solution differences are due simply to the fact that the contour was designed with the HKL throat solution and recomputed here with the source-flow assumption; that is, the contour design is significantly influenced by the assumed throat solution. Had the contour been designed assuming a source flow at the throat, a slightly different contour would have resulted. With these results in hand, the concern arises as to whether any selected contraction contour will actually deliver the throat profile assumed in the design. To the authors' knowledge, this question has never been addressed (or, at least, published).

B. Prior Work An electronic literature search on contraction design for sonic nozzles turned up very few publications. However,

there is a large literature base on design methods for wind tunnel contractions limited to low-speed, incompressible flow. Perhaps the incompressible method most widely referenced is that of Morel.2 Morel's method uses two smoothly joined cubics chosen to provide specified inlet and exit static pressure values and uniform exit flow. Although his development is for low-speed flow, some sonic designs still used his contours. In the area of



compressible contraction flow, Cohen and Nimery3 present a method to design for uniform exit flow at Mach 1. Their method assumes compressible, inviscid, irrotational flow. An axial velocity distribution is specified. Their method provides an analytical series flow solution to generate a contour that will yield uniform sonic flow at the exit. Zannetti4 presents a computational fluid dynamics (CFD)-based method that is a direct solution of the inverse problem of contraction design. He solves the unsteady Euler equations with a moving boundary that drives toward a specified pressure distribution on the wall. Larocca and Zannetti5 later extended this method to rotational flow. Argrow and Emanuel6 present an extensive analysis of the effect of plane-circle contraction geometries on the sonic line in two-dimensional (2D) sharp-throat (minimum length) nozzles. Ho and Emanuel7 present an inviscid design-by-analysis method to design a contraction contour that produces uniform sonic exit flow for application to sharp throat nozzles. It is noteworthy that all of the compressible design methods encountered, perhaps excepting Zannetti, have an objective of designing a contraction contour to yield uniform sonic exit flow. In contrast, the HKL throat solution described above does not yield a uniform sonic flow at the throat.

C. Background In current contraction design practice used at AEDC, the following rules are presently recommended: • The contraction contour must match the geometric throat radius of the supersonic contour. • The contraction contour radius must decrease monotonically all along its length, moving downstream. • The contraction contour should have zero slope at its entrance and where it matches the geometric throat. • The contraction contour should have continuous slope all along its contour. • The contraction contour should have continuous curvature (second derivative) all along its contour. • The contraction contour should match the wall radius of curvature of the supersonic contour at the geometric

throat. • The inflection angle of the contraction contour should be small enough to prevent flow separation. These rules were imparted to the author by James C. Sivells, who developed the internationally known computer

code CONTUR for the design of supersonic nozzle contours (Ref. 1). Sivells also developed the CQCQ contraction model (Ref. 8 gives the mathematical model), which is illustrated

in Fig. 10. The entrance cylinder matches the stilling chamber and would typically be much longer than shown. The cylinder is followed by a quartic polynomial that has zero curvature at both ends and matches the conical section at its upstream end. A second quartic matches the radius of curvature at the throat (determined by the supersonic contour design) and has zero curvature where it matches the conical section. The input to this model is composed of the following quantities:

• The cone half angle (inflection angle) • The length of the entrance cylinder • The overall length of the contraction from entrance to throat • The throat radius • The radius of curvature at the throat Sivells used this model to design the contraction for the Tunnel C Mach 4 "Aerothermal Tunnel" nozzle (Ref. 1

test case). This design used a 30-deg cone half angle on the contraction, which is thought to be a rule of thumb advocated by Sivells as adequate to avoid separation. However, Sivells did base this rule on the separation criterion of Ref. 9.

Sivells’ CQCQ model satisfies the list of design rules, but it has significant limitations. Specifically, if the required contraction length is too small, the model may not yield a smooth continuous contour. The contraction length is a strong function of the cone angle (as well as the throat curvature), but in practice it has been found difficult to increase the angle much beyond about 35 deg while maintaining a continuous contour.

An issue of concern is the throat flow delivered by the contraction design. This is of concern because the MOC-based design codes always assume some specific throat flow profile as a boundary condition on the space-marching numerical solution. It is always implicitly and tacitly assumed (i.e., without verification) that the contraction will deliver the assumed throat flow profile. Since the contraction is designed after the supersonic contour, typically without enforcing rigorous fluid dynamic boundary conditions or the physical flow equations, there appears to be potential for error.

The general approach here is to use a CFD code to evaluate contraction contours based on several different mathematical functions including the CQCQ model. The computations are for the entire nozzle, including both the contraction and the supersonic contour. The focus of the computations is the exit flow uniformity and the effect of the contraction design while holding the supersonic contour fixed. The throat flow profile delivered by the contraction is also examined and compared to the profile assumed by the supersonic contour design code.1



II. Approach An important goal of this effort is to determine whether the contraction does or can deliver the throat profile

assumed by the supersonic contour design code. Since that code is inviscid (although with a post-design boundary-layer correction added on), the present analysis computations are also inviscid. The nozzle selected as a test case is the Tunnel C Mach 4 contour (Fig. 1) designed by Sivells using his code CONTUR.1 Sivells’ original input data set for that (viscous) design is still available, and his contour coordinates have been reproduced as given in Ref. 1. For the present inviscid application, the contour coordinates have been regenerated without the viscous correction. Contraction contours geometrically consistent (to varying degrees) with the inviscid supersonic contour have been computed.

The primary flow solver used here is the Nozzle Analysis Program (NAP) documented in Ref. 10 and referred to herein as CLINE. This program solves the Euler equations in a time-marching fashion using the explicit MacCormack predictor-corrector algorithm. The code is applicable to 2D or axisymmetric nozzle geometries. The equation of state is a thermally and calorically perfect gas defined by the specific-heat ratio and the gas constant. The grid is uniformly spaced and body fitted. The subsonic inlet boundary conditions are based on the space-time characteristics, which rigorously establish exactly what flow properties may (and may not) be specified at the inlet boundary. Here, the total pressure and total temperature are specified to be uniform at the inlet, and the flow angle is zero across the inlet boundary (a vertical line upstream of the throat). Specifically, the axial velocity, the mass flow rate, and the static properties are not specified at the inlet but are computed as part of the solution. The algorithm requires artificial dissipation for stable convergence, and the smoothing parameter is set as recommended in Ref. 10. This results in a small but observable loss of momentum and possibly mass and energy. Accordingly, it is assumed that the losses are similar for all cases run and across the nozzle diameter so that cases may be meaningfully compared.

For analysis with CLINE, the supersonic contour coordinates were interpolated with CONTUR to 0.5-in. stations, and the nozzle exit was slightly truncated at 134 in. from the throat. Sivells’ original CQCQ contraction contour was modified slightly to match the inviscid version of the contour. Based on prior experience with CLINE applications, the contraction inlet was placed at about one inlet diameter upstream of the point where the contour convergence begins. Specifically, the computational inlet was located 86 in. upstream of the geometric throat with the throat arbitrarily set at station zero (Fig. 1). The grid was chosen so that a grid point fell exactly at the throat because this was found to be important for solution accuracy. Every CLINE solution was run with the same number of axial and radial grid points, specifically 441 axial points and 19 radial points. A grid resolution study was not done, but this grid density was previously found adequate for similar nozzles. All solutions were run for 40,000 time steps, which generally decreased the measure of unsteadiness (max |du/dt|) by 3½ orders of magnitude for a quasi-1D initial flow field. The code was compiled with Portland Group Fortran 90 on a Linux workstation running Red Hat Enterprise. The compute time was about a minute.

The MOC analysis code used here (NOZMOCA) is a fairly conventional space-marching MOC code. It is an axisymmetric irrotational MOC code that assumes a thermally and calorically perfect gas. Several throat flow models are provided, including the HKL solution and radial source flow. The characteristic equations are solved by a simple fixed-point iteration. The solution is constructed on solution surfaces that are roughly axis-normal but which contort as the solution marches downstream. NOZMOCA was written explicitly as part of a multithroat contour design capability. A complementary contour design code NOZMOCD was also developed. These codes may be the subject of a future paper.

Several contraction contour models were tested. The CQCQ model (Fig. 10), which is documented in Ref. 8, was applied over a small range of cone angles (25, 30, and 35 deg) while maintaining the other boundary conditions.

To examine a larger range of contraction angles, a function based on sinn was developed.

( ) ( )[ ]χζ−−+= 1121 yyyy

2

112

1 πχ ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−

≡xxxx

( ) χχζ nsin≡



where (x1,y1) is the inlet and (x2,y2) is the throat. Control over the inflection angle is indirect through the contraction length or the exponent n. Only n = 2 was tested here. Cases for n > 2 may be of interest, but for these cases y2" = 0. Note that at both x1 and x2, y' = 0. In general, the throat curvature is not matched to that of the supersonic contour.

To examine the effect of curvature discontinuities at other than the throat, a double ellipse model was developed (Fig. 11). This model is composed of two ellipses that are tangent at their match point. The model is as follows. (The equation order given below is also the computational sequence, i.e., variables as defined in Fig. 11).

( )

( )2

2

332

32

323

1 ⎥⎦

⎤⎢⎣

⎡′′′

−−

′′−=

yyhx

yhxb

333 byk +=

3

33 y

ba′′

=

2

3

32332 1 ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−−=

ahxbky

( )( ) ( )21212

22

212212

1 2 yyyhxyyyyhxk

−+′−−+′−

=

111 kyb −=

2

1

12

121

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

−=

bky

hxa

In this model (x1,y1) is the inlet, (x3,y3) is the throat, and (x2,y2) is the tangency point of the two ellipses. The variables a and b are the ellipse major and minor axes, respectively, with subscript 1 corresponding to the entrance ellipse, and subscript 3 to the throat ellipse. The variables h and k are the coordinates of the ellipse center. The model input quantities are the variables x1, y1, x3, y3, y3", x2, and y2'.

III. Results Figures 12 through 14 show the contraction contours selected for evaluation. Figure 12 shows three of Sivells’

CQCQ contours for different cone angles, 25, 30, and 35 deg. Each of these contours matches the wall radius of curvature of the supersonic contour at the throat. Attempts to obtain a useful contour with a 40-deg angle were unsuccessful. Of the contours to be considered here, the CQCQ model is the best at satisfying the previous list of desired contour traits. The 30-deg CQCQ contour will be considered the baseline case against which all others are evaluated. Figure 13 shows the contours for the sin2 model. Contours for contraction angles from 31 to 60 deg were created. However, this analytical model was not developed to the point where the inflection angle could be specified; accordingly, the angles given (Fig. 13) are approximate and are rounded to the nearest whole degree. Note that the 31-deg case matches the baseline very closely. However, none of the sin2 contours matches the throat radius of curvature, although the 31-deg case is close (Table 1). Figure 14 shows two double ellipse contractions selected for analysis. These contractions have inflection angles of 38 and 60 deg. A suitable contour could not be found for smaller angles. Both ellipse contractions match the throat radius of curvature, but both also have large curvature



discontinuities at the tangency point (point 2 in Fig. 11). The throat radius of curvature of each contraction is shown in Table 1.

Table 1. Contraction curvature at throat.

Function Inflection

angle, deg

y", /in. 1/y", in. Radius ratio*

ellipse 60 4.4531E-02 2.2456E+01 6.0000E+00 ellipse 38 4.4531E-02 2.2456E+01 6.0000E+00 CQCQ 30 4.4531E-02 2.2456E+01 6.0000E+00 CQCQ 25 4.4531E-02 2.2456E+01 6.0000E+00 CQCQ 35 4.4531E-02 2.2456E+01 6.0000E+00

sin2 31 4.7057E-02 2.1251E+01 5.6779E+00 sin2 42 1.0328E-01 9.6824E+00 2.5870E+00 sin2 50 1.8823E-01 5.3127E+00 1.4195E+00 sin2 55 2.6052E-01 3.8384E+00 1.0256E+00 sin2 60 3.8414E-01 2.6032E+00 6.9554E-01

*Radius ratio = ratio of wall radius of curvature at throat to throat radius Throat radius = 3.7427 in. Entry in bold is baseline.

Initially, a few inviscid solutions were computed for validation purposes for the CQCQ contraction using the

unstructured code USM3D.11, ‡ Figures 15 and 16 show USM3D results for the 25-deg CQCQ contraction. Figure 15 shows the Mach number profile at the nozzle exit for both USM3D and CLINE. The design Mach number (CONTUR) is Mach 4, but this is an input boundary condition to CONTUR and not a computed profile as possibly implied by Fig. 15. Undeniably, both USM3D and CLINE show losses compared to the Mach 4 target number. Also shown is a computed profile from the new MOC code NOZMOCA, which is much closer to design than either of the two more conventional CFD codes. The conclusion, not unexpected, is that both USM3D and CLINE are dissipative compared to MOC. In any event, however, both CFD codes are within about 0.5 percent of the target Mach number. This result is regarded as validation of the CLINE code, with the CLINE code being observably (perhaps not significantly) better than the USM3D code for the present case. Figure 16 compares the sonic line from USM3D with the HKL analytical inviscid solution for a circular arc throat. The slight noise in the USM3D profile is probably related to the unstructured grid and the need to interpolate the flow solution. Regardless, USM3D and HKL are in good agreement, and this result validates USM3D for the present application. CLINE results are not shown here since a convenient procedure for sonic line extraction is not coded in the CLINE software.

Next, flow solutions are presented for the three CQCQ contractions with inflection angles of 25, 30, and 35 deg. Figures 17 through 23 are applicable. Figure 17 compares the Mach number profiles from CLINE at the geometric throat for the three CQCQ contractions with the HKL solution. Clearly, to excellent engineering accuracy, all results are in agreement. There is no apparent effect of contraction angle on the Mach number profile at this point. This result is a primary finding and verifies that Sivells’ CQCQ contraction model delivers the HKL-based throat flow assumed by his design code. Figures 18 through 23 give the CLINE profiles at the nozzle exit. The profiles include the Mach number (Fig. 18), the static pressure (Fig. 19), the static density (Fig. 20), the static temperature (Fig. 21), the axial velocity component (Fig. 22), and the flow angle (Fig. 23). For the thermodynamic properties and velocity, each flow profile is presented as percent deviation from the average of the profile extremes (maximum and minimum values) for each solution. The goal here is to compare the 25- and 35-deg inflection angles with the baseline 30-deg case. The greatest deviation from uniformity is less than 0.06 percent for static pressure, and all three contractions have very similar results. A second finding, therefore, is that small deviations (±5 deg) in inflection angle are not significant.

Next, the flow solutions for five sin2 contractions are compared with the CQCQ 30-deg baseline solution. Figures 24 through 30 are applicable. Figure 24 compares the throat Mach number profile of the five sin2 contractions with the CQCQ 30-deg baseline and the HKL solution. In contrast to the two other CQCQ cases, the ‡ USM3D solutions were computed by Moufid Aboulmouna, AEDC/ATA.



sin2 contractions have a significant effect on the throat flow. Note, however, that the 31-deg sin2 contraction, which matches the 30-deg CQCQ contour fairly well (Fig. 13), yields the same throat profile as the CQCQ contraction. The issue then is how the deviations of the other contraction angles affect the exit flow uniformity. Figures 25 through 30 follow the same scaling procedure as do the CQCQ profiles. The exit Mach number profiles (Fig. 25) show a maximum deviation from the baseline of about 0.012 for the 60-deg inflection angle. In terms of real test facilities, this is not a very large deviation, but it translates into ±0.5 percent for static pressure (Fig. 26), which would be measurable in a good aerodynamic facility. The density, temperature, and velocity (Figs. 27-29) all show considerably less deviation. The maximum flow angularity (Fig. 30) is less than 0.1 deg for the 60-deg sin2 contraction. The primary finding here is that large deviations in inflection angle and throat curvature discontinuity have a measurable effect on static pressure. However, it is unclear at this point whether only one or both of these parameters are important.

To help resolve this question, the double ellipse contraction model was applied. The model accommodates large inflection angles while still matching the throat curvature. It also introduces large curvature discontinuity at the ellipse-ellipse tangent point. Two solutions were computed with inflection angles of 38 and 60 deg (a 30-deg contour could not be obtained). Figures 31 through 37 give these results. Clearly, the throat flow profiles (Fig. 31) for the double ellipse are as good as those for the baseline. The exit profiles (Figs. 32 through 37) show little more deviation from the baseline than does the CQCQ contraction, even though one of the ellipse contractions has an inflection angle of 60 deg. Thus, a major finding here is that a contraction contour that matches the throat curvature will permit nozzle exit flow uniformity almost as good as the contraction contour assumed in the design of the supersonic contour. Furthermore, this finding appears to be a valid conclusion even if there are large contour differences upstream of the throat, including slope differences and curvature discontinuities. However, the issue of flow separation is not addressed and might be affected by the contraction contours examined here.

IV. Critique This section is intended to document deficiencies in the present analysis. First, the study addressed a limited

range of contraction geometries. Axisymmetric geometries were considered, but not 2D geometries. Slope discontinuities, which are present in arc heater nozzles, were not addressed. Contractions with fully continuous curvature but large inflection angles were not considered primarily because a suitable set of functions has not been found. Submerged bellmouths and nonaxisymmetric configurations have not been addressed. Also, only a single supersonic contour was considered, that of a high-quality aerodynamic nozzle for Mach 4. In contrast, nozzles for the AEDC arcs and for APTU (AEDC Aerodynamic and Propulsion Test Unit) are very short by comparison, and their flow may respond differently to contraction effects. Similarly, nozzles with significantly larger or smaller exit area ratios may produce flows that respond differently from the present Mach 4 nozzle. The assumption of thermally and calorically perfect gas thermodynamics limits the applicability of the present findings to relatively low-temperature and low-pressure facilities; thus, the findings may not be valid for arc heaters, combustion-heated facilities, or shock tunnels. Even Tunnel C is known to have deviations from simple thermodynamics. However, this is an assumption of the contour design procedure. The inviscid assumption is appropriate as long as MOC-based design codes are relevant, but there are likely significant viscous effects (e.g., flow separation) that should be considered. Though results were not presented, the 60-deg sin2 contraction was predicted to separate near the throat using a semi-empirical criterion,9 a condition that might be catastrophic for flow quality. Another deficiency is the assumption that the contraction inlet has uniform flow. This is not a necessary assumption with CLINE, but stilling chamber flow profiles are rarely measured. However, simple analysis shows that velocity fluctuations in the stilling chamber, where the kinetic energy is small, are insignificant in the nozzle exit flow where much of the internal energy has been converted to kinetic energy. With low flow speed in the stilling chamber, gradients in total or static pressure cannot be generated unless strong vortices (as associated with bellmouth inlets) are present. However, total temperature or enthalpy gradients present in the stilling chamber will be preserved in the test section flow. None of these possibilities has been considered here. CFD specialists will charge that the CLINE flow solver is obsolete. The author would respond that CLINE shows less dissipation than a modern unstructured solver (USM3D), though neither is as good as the relatively ancient MOC. An acoustic-capable solver might yield results comparable to MOC, but none is currently operational at AEDC.



V. Conclusions and Recommendations The following conclusions are proposed based on the previous material: • The cylinder-quartic-cone-quartic contraction model developed by Sivells very accurately delivers the throat

flow profile expected by his MOC design code. • Deviations in contraction inflection angle of 5 deg, while throat curvature is continuous, do not have a

significant effect on either throat flow or exit flow. • Large variations in the contraction inflection angle do not significantly affect exit flow uniformity if the

contraction contour has continuous curvature at the throat (flow separation was not considered here). • Curvature discontinuities at the throat cause measurable deviation from the expected exit flow and alter the

throat flow significantly. The general conclusion is that if the throat-region curvature is continuous from the contraction contour to the

supersonic contour, other factors such as the contraction inflection angle or upstream curvature discontinuities do not have much effect on the inviscid flow uniformity at the nozzle exit. Therefore, the traditional requirement for curvature continuity upstream of the throat is not supported.

Based on results herein, there is not sufficient evidence to justify development of an aerodynamically rigorous procedure for contraction design. However, the issue should be revisited if a nozzle design procedure is developed that relaxes some of the constraints in the current MOC procedure.

Acknowledgments The work reported herein was conducted by the Arnold Engineering Development Center (AEDC), Air Force

Materiel Command (AFMC), at the request of the Capabilities Integration Directorate, Technology Division (AEDC/XRS). The results of the technology development effort were obtained by the Aerospace Testing Alliance (ATA), the operations, maintenance, information management, and support contractor for AEDC, AFMC, Arnold Air Force Base, Tennessee, under AEDC Project Number 11808, Modeling, Simulation and Analysis. The ATA Project Manager was Ms. B. D. Heikkinen, and the AEDC Air Force Project Manager was Mr. Jeffrey T. Staines. The report was peer reviewed by Dr. E. S. Powell and Mr. W. E. Milam, ATA.

References 1Sivells, J. C., “A Computer Program for the Aerodynamic Design of Axisymmetric and Planar Nozzles for Supersonic and

Hypersonic Wind Tunnels,” AEDC-TR-78-63, December 1978. 2Morel, T., “Comprehensive Design of Axisymmetric Wind Tunnel Contractions,” Journal of Fluid Engineering

(Transactions of the ASME), June 1975, pp. 225-233. 3Cohen, M. J., and Nimery, D. A., “The Design of Axi-Symmetric Convergent Cones with Plane Sonic Outlets,” Israel

Journal of Technology, Vol. 10, No. 1-2, 1972, pp. 69-83. 4L. Zannetti, L. “Time-Dependent Method to Solve the Inverse Problem for Internal Flows,” AIAA Journal, Vol. 18, No. 7,

Article No. 79-0013r, July 1980, pp. 754-758. 5Larocca, F., and L. Zannetti, L., “Design Method for 2-D Transonic Rotational Flows,” AIAA 95-0648, 33rd Aerospace

Sciences Meeting and Exhibit, Reno, NV, Jan. 9-12, 1995. 6Argrow, B. M., and Emanuel, G., “Computational Analysis of the Transonic Flow Field of Two-Dimensional Minimum

Length Nozzles,” Journal of Fluids Engineering, Vol. 113, Sep. 1991, pp. 479-488. 7Ho, T.-L., and Emanuel, G., “Design of a Nozzle Contraction for Uniform Sonic Throat Flow,” (Technical Note) AIAA

Journal, Vol. 38, No. 4, Apr. 2000, pp.720-723. 8Shope, F. L., “Contour Design Techniques for Super/Hypersonic Wind Tunnel Nozzles,” AIAA-2006-3665, 24th AIAA

Applied Aerodynamics Conference, San Francisco, CA, June 5-8, 2006. 9Stratford, B. S., “The Prediction of Separation of the Turbulent Boundary Layer,” Journal of Fluid Mechanics, Vol. 5, 1959,

pp. 1-16. 10Cline, M. C., “NAP: A Computer Program for the Computation of Two-Dimensional, Time-Dependent, Inviscid Nozzle

Flow,” Los Alamos Scientific Laboratory LA-5984, January 1977. 11Frink, N. T., “Tetrahedral Unstructured Navier-Stokes Method for Turbulent Flows,” AIAA Journal, Vol. 36, No. 11,

November 1998, pp. 1975-1982.



Figure 1. AEDC Tunnel C Mach 4 “Aerothermal Tunnel” nozzle.

Ratio of throat wall radius of curvature to throat radius = 6

Throat

Figure 2. Throat flow sonic line and throat characteristic from HKL transonic solution for AEDC Tunnel C Mach nozzle.



Start lines approximately 1 in. downstream of throat

Figure 3. Starting flow profiles for motivational MOC solutions.

Figure 4. Characteristic net with HKL start line.



Figure 5. Characteristic net in near-throat region with HKL start line.

Figure 6. Characteristic net with source-flow start line.



Figure 7. Characteristic net in near-throat region with source-flow throat start line.

Note perturbationsNote perturbations

Figure 8. Centerline static pressure distributions with HKL and source-flow start lines.



Figure 9. Exit Mach number profiles for HKL and source-flow start lines.

flowflow

CLCL

cylinder

quartic

quartic

cone

geometric throat

Figure 10. Sivells’ cylinder-quartic-cone-quartic (CQCQ) contraction.



flow

CL

(x1,y1)

(x2,y2)

(x3,y3)

(h3,k3)

(h1,k1)

b1a1

b3a3

Given: x1, y1, x3, y3, y3”, x2, y2’N.b.: Ellipses are not necessarily identical

flowflow

CLCL

(x1,y1)

(x2,y2)

(x3,y3)

(h3,k3)

(h1,k1)

b1a1

b3a3

Given: x1, y1, x3, y3, y3”, x2, y2’N.b.: Ellipses are not necessarily identical

Figure 11. Double ellipse contraction.

Figure 12. Contraction contours using Sivells’ cylinder-quartic-cone-quartic function.



Figure 13. Contraction contours using sin2 function.

Figure 14. Contraction contours using double ellipse.



Figure 15. Exit Mach number profile from CLINE and USM3D.

Figure 16. Sonic lines from USM3D and the HKL analytical solution.



CLINE code

Sivells’ cylinder-quartic-cone-quartic function (CQCQ)

Figure 17. Throat Mach number profiles from CLINE compared to HKL solution.


CLINE code

Figure 18. Exit Mach number profiles for three CQCQ contractions.



Figure 19. Exit static pressure profiles for three CQCQ contractions.

Figure 20. Exit static density profiles for three CQCQ contractions.



Figure 21. Exit static temperature profiles for three CQCQ contractions.

Figure 22. Exit axial velocity profiles for three CQCQ contractions.




CLINE code

Figure 23. Exit flow angle profiles for three CQCQ contractions.

CQCQ is baseline

CLINE code

Figure 24. Throat Mach number profiles for sin2 contractions.



CQCQ is baseline

CLINE code

Figure 25. Exit Mach number profiles for sin2 contractions.

Figure 26. Exit static pressure profiles for sin2 contractions.



Figure 27. Exit static density profiles for sin2 contractions.

Figure 28. Exit static temperature profiles for sin2 contractions.



Figure 29. Exit axial velocity profiles for sin2 contractions.

CQCQ is baseline

CLINE code

Figure 30. Exit flow angle profiles for sin2 contractions.



CLINE code

Figure 31. Throat Mach number profiles for double ellipse contractions.

CLINE code

Figure 32. Exit Mach number profiles for double ellipse contractions.



Figure 33. Exit static pressure profiles for double ellipse contractions.

Figure 34. Exit static density profiles for double ellipse contractions.



Figure 35. Exit static temperature profiles for double ellipse contractions.

Figure 36. Exit axial velocity profiles for double ellipse contractions.



CLINE code

Figure 37. Exit flow angle profiles for double ellipse contractions.

[american institute of aeronautics and astronautics 26th aiaa aerodynamic measurement technology and...

Documents