Integrated supersonic wind tunnel nozzle

Junmou SHEN, Jingng DONG, Ruiqu LI, Jing ZHANG, Xing CHEN,Yongming QIN, Hndong MA

a China Academy of Aerospace Aerodynamics, Beijing 100074, China

b School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M1 7DN, UK

KEYWORDS

AbstractIn supersonic wind tunnels, the airflow at the exit of a convergent-divergent nozzle is affected by the connection between the nozzle and test section, because the connection is a source of disturbance for supersonic flow and the source of disturbance generated by this disturbance propagates downstream. In order to avoid the disturbance, the test can only be carried out in the rhombus area.However,for the supersonic nozzle,the rhombus region is small,limiting the size and attitude angle of the test model.An integrated supersonic nozzle is a nozzle and a test section as a whole, which is designed to weaken or eliminate the disturbance. The inviscid contour of the supersonic nozzle is based on the method of characteristics. A new curve is formed by the smooth connection between the inviscid contour and test section, and the boundary layer is corrected for the overall curve. Integrated supersonic nozzles with Mach number 1.5 and 2 are designed, which are based on this method. The flow field is validated by numerical and experimental results. The results of the study highlight the importance of the connection about the nozzle outlet and test section.They clearly show that the wave system does not exist at the exit of the supersonic nozzle,and the flow field is uniform throughout the test section.

1. Introduction

For supersonic wind tunnels, the connection between nozzle and test section is usually a convex surface or backward step,and the discontinuity of the connection brings the disturbance to the test section. The disturbance behavior coming from the supersonic nozzle outlet is based on the pressure ratio of the nozzle outlet static pressure to the back pressure. When the supersonic nozzle is operated at the pressure ratio below one,the oblique shock waves are formed from the nozzle outlet even inside the nozzle. However, for the normal operation of the supersonic wind tunnel, the pressure ratio less than one is unlikely to happen, so this paper only discusses the case where the pressure ratio is greater than one. If the pressure ratio is greater than one, the supersonic flow through the nozzle outlet negotiating the convex surface or backward step undergoes a Prandtl-Meyer expansion fan.1The Prandtl-Meyer expansion fan interacts at the central zone of the test section and then reflects on the test section wall or the boundary of the free jet. The Mach number downstream of the expansion fan is increased, while pressure, density, and temperature are decreased, making it unsuitable for the test in the area. The Mach angle of the first expansion wave is large in supersonic flow, resulting in a small test rhombus region.Not only is the size of the test model limited, but its attitude angle is also restricted. Thus, it is necessary to enlarge the test area.

The design method of the nozzle is mainly divided into direct design method and design by analysis.The direct design method is easier than design by analysis,because the boundary conditions have easier access to apply.For the supersonic wind tunnel nozzle, the proper direct design method can deliver quite good flow field and is adopted by this paper. The direct design method based on the Method Of Characteristics(MOC)is used to design the inviscid contour of the supersonic nozzle,which was originally proposed by Prandtl and Busemann.2Foelsch3assumed that there was a source flow in the nozzle,considered and compared the air condition in the conical nozzle, and obtained the analytical expression of the nozzle line.Cresci4inherited the idea of source flow and obtained a continuous curvature of the nozzle by setting a transition region between the source flow and the uniform region. Sivells5accepted and developed the transition region, completely obtained the inviscid contour,and improved the nozzle design theory based on the method of characteristics. To correct the inviscid contour for the viscous effects, the Boundary Layer(BL) calculation based on von Karman integral momentum equation is essential.6

The test section and nozzle need smooth connection,which can reduce the disturbance caused by the discontinuity. The Integrated Supersonic Nozzle (ISN) is a nozzle and a test section as a whole, which is designed to weaken or eliminate the disturbance.The present paper performs the design method of the ISN and the numerical computation and wind tunnel experiment for them,and the ISN has been set up in the China Academy of Aerospace Aerodynamics (CAAA). The ISN is designed and manufactured according to the industrial standards. There is a two-dimensional nozzle, where two opposite walls are contoured in a convergent-divergent shape but are bounded by parallel walls, providing a rectangular crosssection at each site.

The Mach number is changed less than 0.01 in the 1.6 m long test section, and aerodynamic forces and moments of the AGARD-B model in the test section agree well with the reference data.7-9There is no expansion wave at the nozzle outlet and test section, so that the size and attitude angle of the test model can be enlarged appropriately.

2. Design method of ISN

2.1. Expansion section

The ISN consists of a supersonic nozzle and test section. The design techniques of the supersonic nozzle smooth MOCbased inviscid contour plus BL correction.The design method is based on Sivells.5,6The basic idea is that the specification of a uniform nozzle outlet flow along with three parts of centerline Mach number distribution, changing monotonically from sonic flow at the throat to the nozzle exit Mach number as illustrated in Fig.1. In the present method, the downstream of the throat is the left-running characteristics region, the radial flow region, the right-running characteristics region,and test region.

In the present method,throat characteristic IT is combined with the transonic solution of Hall,10,11and the center velocity distribution from sonic point I to point E is described by a third-degree polynomial. Characteristics GE and AB belong to the radial flow region and the flow region can be accurately described by the radial flow equation.5From the point B to point C, Mach number distribution BC is specified by a fourth-degree polynomial. The exit characteristic CD is a straight line because it is defined by the exit Mach number.When the boundary conditions of MOC regions are defined,the characteristic grid point along TIEG and ABCD could be computed in Fig.1. In a supersonic flow, two characteristic lines from every point, and the velocity of every point is u,which are a left running characteristic(S+)and a right running characteristic(S-),are shown in Fig.2.Characteristic lines are Mach lines, which are normally curved. The S+characteristic has an angle θ+μ, and the S-characteristic has an angle θ-μ,where θ is the flow angle and μ is the local Mach angle.

All points on a characteristic line have the same value of S+,-. In Fig.2, points 1 and 2 locate the same S+and points 2 and 3 lie on the same S-characteristic

where Ma is Mach number,ν is the Prandtl-Meyer angle given by

where γ is specific heat ratio.If the coordinates,flow angle and Mach number are known at primary points 1 and 2,the equations of the characteristics line, i.e.

The coordinates of point 3 are the intersection of lines(S+)2and(S-)1,and the coordinates of point 4 are the intersection of lines(S+)1and(S-)2.The network of the region ABCD is completed by right-running characteristics. The process in region TIEG is similarly calculated by the left-running characteristics. The TG segment and AD segment are determined according to the principle of the mass of conservation, where the mass flow through each of the left running characteristics or the right running characteristics is equal.The contour from point G to point A is a straight line. Thus the inviscid contour TD is determined.

Fig.1 Symbol and design procedure based on Sivells.

Fig.2 Characteristic line.

The actual flow is viscous, resulting in a certain angle between the nozzle outlet flow and the centerline.If the downstream of the nozzle outlet is inconsistent with the angle,there will be the disturbance.The supersonic flow passes through the wall at an angle to the direction of the flow, and small disturbances created by discontinuous boundary will propagate away.The body shape for generating Mach waves is a concave corner, which generates an oblique shock, or a convex corner,generating an expansion fan,φ is corner angle,as illustrated in Fig.3(a)and(b).Where Ma,ρ,p,h,p0are Mach number,density, static pressure, enthalpy, and total pressure, respectively.In order to reduce or eliminate small disturbances,the inviscid contour of nozzle outlet, and test section must be smoothly connected. The inviscid contour test section is a straight line,whose slope is equal to the slope of the nozzle outlet.

To obtain the viscid contour of the nozzle and test section,each ordinate point of the inviscid contour needs to add a correction for the BL growth δ*. The method of calculating the BL correction is employed by the von Karman integral momentum equation for planar flow, assuming a power law velocity profile with the BL. The BL correction is made for a design condition of the stagnation pressure and temperature.The von Karman momentum is written in the form

Fig.3 Supersonic flow passes through wall.

where Θ is momentum thickness, and the value Ma, yw, x are Mach number, y-coordinate and x-coordinate respectively.The term β=arctan(dyw/dx) can be obtained from the inviscid contour. For the wind tunnel nozzles, the form factor H6and the skin friction coefficient Cf6are as a function of Reynolds (Re) number based on the momentum thickness Θ. It is convenient for Eq.(8)to be solved numerically by the applications of the parabolic technique of integration.Note that the integration is started at the nozzle throat.

For a two-dimensional nozzle with a pair of parallel side walls, there is the BL growth on four walls. The BL growth on the parallel walls is not corrected directly, but the correction on the parallel walls is incorporated into the contour nozzle walls. The displacement thickness of the effective BL is transformed from the correction of parallel and contour walls(see Figs. 4(a) and (b))

where δcis the BL displacement thickness of contour nozzle,y is ordinate of the inviscid nozzle contour, and W is the width of the parallel side walls.

2.2. Contraction section

The nozzle contraction section is designed by using the optimized Witoszynski curve,12which is determined by means of shifting the axis of the Witoszynski curve. The Witoszynski curve equation is described as

where y*and y0are the half height of the throat and subsonic contraction respectively, x is the length from the inlet of the contraction section,and L is the length of the contraction section. At this time, the method of shifting the axis can be used to optimize the contraction section curve, that is, setting R1and R2to represent the actual half heights of the throat and subsonic contraction

where Rhis the amount of the displacement axis, and the actual curve coordinates are calculated by replacing y* and y0with R1and R2,and subtracting all the coordinates from Rh.

2.3. Contour smoothing

For the convenience of numerical calculation and design processing,the nozzle contour based on the MOC and BL correction needs to be smoothed using Hermit Interpolation Methods (HIM). HIM can be adjusted to provide a good fit for the coordinates.13

2.4. Contour optimistic

Conventional nozzle contour is obtained by the MOC method.According to the Sivells’ method, the Mach number at points B and C, and at points E, G and A are related by

where v is the Prantdtl-Meyer angle,and θ is the flow angle relative to the nozzle centerline.

Obviously, if point A and point G coincide

The slope of the nozzle outlet is larger than 0.If the nozzle and the test section are only directly connected,the airflow will continue to expand in the test section,and the test airflow will be disturbed. In the actual design process, in order to ensure the uniform flow field of the test section,the integrated nozzle contour needs to be iterated and optimized. The flow field at the nozzle outlet is under-expansion and continues to expand in the test section. This process is achieved by adjusting the Mach number of point E and point B, and the underexpansion factors λ1and λ2are introduced.

Fig.4 Schematic of BL thickness.

Fig.5 Nozzle contour and contour slope.

The factors λ1and λ2are determined by the height of the nozzle outlet and the maximum expansion angle,and the range of factors λ1and λ2are from 0.8 to 1. The Mach number distribution from point B to point C with the criteria must be taken to ensure that the first derivative is monotonic and the second derivative is negative.With the increase of the designed Mach number, the value can be appropriately increased.

Geometrically, reducing the disturbance, the smooth connection of the nozzle contour and the test section is necessary.Physically,when the boundary layer growth amount of the test section is matched with the test airflow that continues to expand in the test section,the Mach number of the test section can be very uniform.The inviscid contour of the test section is a straight line, which is tangent to the inviscid contour of the nozzle outlet. Similar to the inviscid nozzle contour, the boundary layer correction is also based on von Karman integral momentum equation. The aerodynamics parameters of the inviscid test section is the same as those as the inviscid nozzle outlet. The effect of the under-expansion factors λ1and λ2are to allow the nozzle outlet airflow to continue to expand slightly downstream, and there are not small disturbances in the test section.

Fig.6 Optimization process.

The general iteration and optimization process established to determine the ISN contour is outlined in Fig.6. When the Mach number index of the test section meets the excellent standard,14the optimization is completed, and the number of iterative steps is usually less than 10.

3. Numerical methods

The compressible Navier-Stokes equations are utilized for the 2-D and 3-D viscous cases. The Reynolds-averaged governing equations with the Spalart-Allmaras turbulence model15are discretized by cell-centered finite volume method. Jameson and Baker’s four stage Runge-Kutta method16is used for time integration. Second upwind difference schemes17,18with a variety of higher order limiters are employed to calculate the convective terms in the transport equations. Fully implicit time integration schemes are available for steady flow simulation. The MUSCL scheme19with the van Albada flux limiter is adopted for the third order accurate numerical solution.

For each nozzle, 2-D and 3-D models are used for calculation and each mesh system is the same in the different nozzles.The structured grids are employed by this calculation and are presented in Figs.7(a)and(b)for the 2-D and 3-D case.In the 2-D application, a mesh system of 1001×201 points is employed by Δxmin=1×10-4m located at the throat region around, and Δymin=1×10-5m at the wall. And applied in the 3-D case, a mesh system of 1001×201×201 points is employed, x-direction and y-direction grid distribution are the same as 2-D case, and Δzmin=1×10-5m at the wall.The grid is employed for the 2-D and 3-D case in steady turbulent calculation, and y+ value is less than one.20

17. Came to a meadow: According to Miller, to dream of meadows, predicts happy reunions under bright promises of future prosperity (377).Return to place in story.

According to the actual flow conditions,the initial pressure and temperature are chosen from the data obtained by the nozzle calibration test.The inlet pressure of nozzle Ma=1.5 and Ma=2 is 0.15 MPa and 0.186 MPa, respectively. The inlet temperature of both nozzles is 287 K.

4. Experimental setup

4.1. Pressure rake

Fig.7 Calculation model of nozzle.

Fig.8 Pressure rake.

Two integrated supersonic wind tunnel nozzle was developed at the end of November 2017. The nozzle inner walls of the nozzle are machined with high precision.To calibrate the flow field, a modular cross arm calibration rake is designed and manufactured, as illustrated in Fig.8(a).21Each pressure sensor has a distance of 35 mm on the rake and 26 sensors are installed on one side. In addition to a sensor on the center axis point, a total of 53 sensors are installed. The calibrated cross-section is 910 mm×910 mm, accounting for about 75.6% of the test section.

Pressure probe tips are cylindrical with outside/inside diameters respectively of pitot probe 2 mm/0.8 mm. Errors in pitot pressure due to pressure transducer error are±0.05% Full-Scale (FS). Facility unsteadiness and setpoint errors are less than transducer errors for pressures.When calibrating the flow field, the rake is measured every 50 mm along the direction from the nozzle outlet to test section outlet. Fig.8(b)21shows the pressure rake installed in the test section. The test data in the x-oriented direction covers 1.6 m. Each flow field calibration test is repeated 7 times.

Fig.9 AGARD-B model.

4.2. Calibration model

The aerodynamic forces and moments are measured using the AGARD-B model6in the test section, which is tested on N648EA multi-piece internal six-component balance.AGARD-B model is a body-wing configuration. All its dimensions are given in terms of the body diameter ‘‘D”, so that the model can be produced in any scale, as appropriate for a particular wind tunnel. In the present paper, AGARDB model is the slenderness ratio of 8.5, D=102 mm in diameter and the relative thickness of 0.04, and G(x,y,z) is the coordinate of gravity center. Airfoil is symmetric circular arc form, as shown in Fig.9(a).7Fig.9(b) shows AGARD-B model installed in the test section. Each flow field calibration test is repeated 7 times.

Fig.10 Numerical calculation of Mach number in original nozzle and test section.

Fig.11 Numerical calculation of Mach number contour in ISN.

Fig.12 Experimental results of Mach number contour in ISN.

5. Presentation of results

5.1. Mach number

2-D results of the different calculation Mach number contours about the original and optimistic nozzle and test section are shown in Figs. 10(a)-(b) and Figs. 11(a)-(b). Figs. 10(a)and (b) are based on the direct smooth connection condition of the nozzle and the test section. It can be clearly seen that the expansion wave system exists in the test section and the test area is reduced. Fig.11(a) and Fig.11(b) show the calculation results of the ISN. It’s observed that the flow field is quite uniform, and there is no shock wave and expansion in the nozzle exit and test section. 3-D results of the center cross section are very similar to 2-D results, and the centerline data and nozzle outlet data are given as following in detail.

In the typical experiment, the flow field of the test section is measured by pitot pressure sensor, and the Mach number is calculated from pitot pressure, the total pressure and the total temperature in the test section. The isentropic expansion is supposed in the nozzle, and then the isentropic equations are used to calculate Mach number.22Flow field survey data indicate the spatial variation of the local Mach number is nominally ±0.003. The cross center area of Mach number contours is shown in Figs. 12(a)-(d). Figs. 12(a) and (c) show the Mach contours in the center contour walls, and the Mach number contours of the center parallel walls are shown in Figs. 12(b) and (d). The long length of nearly constant Mach number in the test section is apparent, and the flow field is quite uniform. In the actual flow process, due to processing and installation reasons, the small favorable pressure gradient causes local disturbance of the flow field in the Ma=1.5 test section (see Figs. 12(a) and (b)). However,the disturbance area is small and has little effect on the experimental area.

Figs. 13(a) and (b) show the comparison of the Mach number in the nozzle contour outlet of the center cross section under 2-D, 3-D and experiment condition. Note that the results of 2-D, 3-D, and experiment match well. The initial conditions of calculation and experiment cannot be exactly the same due to the presence of components such as the stable section in the upstream of the wind tunnel nozzle, and there are certain deviations in calculation and experiment. The results of the 2-D calculation are larger than those of the 3-D numerical calculation under some initial and boundary condition.This is because the parallel walls of the nozzle affect the flow field.

Figs.14(a)and(b)show the x-oriented vorticity contour in the nozzle outlet cross section (x=4.8 m). Strong vortices Ω exist near the parallel walls and the corners, which affect the flow field near the contour parallel.For the center contour line of the nozzle outlet, the BL thickness of 2-D and 3-D numerical results in Ma=1.5 nozzle are 0.054 m and 0.072 m,whose results in Ma=2 nozzle are 0.059 m and 0.083 m, respectively.However,the influence of the parallel walls and the corners on the flow field is very small and is not in the uniform area.The vorticity of Ma=2 has a larger influence range than the vorticity of Ma=1.5.

Fig.13 Distribution of Mach number in nozzle outlet of center cross section.

Fig.14 x-oriented vorticity contour.

The comparison of Mach number in the nozzle centerline is illustrated in Figs. 15(a) and (b). Note that the results of 2-D,3-D and experiment also match well. Similar to the results at the nozzle outlet, the test results are between the 2-D and 3-D calculation results.

Tables 1 and 2 give some parameters of the centerline in the contour wall under the whole test section,where Ma-is average Mach number, ΔMa| |maxis the maximum Mach number deviation, and δMais Mach number variance. δMais one of the most important indicators of the nozzle flow field, and δMaincluding Ma=1.5 and Ma=2 is better than the reference excellent indicators140.006 and 0.007 respectively.

5.2. Aerodynamic forces and moments

Analysis of the symmetry of the AGARD-B model in ISN is done by measuring aerodynamic coefficients of the symmetrical aerodynamic configurations. The angle of attack α of the AGARD-B model is varied from -10° to 10°, and the experimental data are compared with the Refs. 7-9, as shown in Figs.16(a)-(d).The Cn～α curves and CM～α curves show that the experimental data in ISN are in good agreement with the reference data,where Cnis the normal force coefficient,and CMis pitching moment coefficient.In general,the flow field uniformity in the ISN meets the requirements of normal wind tunnel tests.

Fig.15 Mach number line in centerline of nozzle contour wall.

Table 1 Flow field parameters of Ma=1.5.

Table 2 Flow field parameters of Ma=2.

Fig.16 Comparison between experimental data and reference data of AGARD-B model.

6. Conclusions

This work demonstrates the MOC and BL method of the ISN design.The flow field quality of the ISN has been studied.Both computational methods and experimental data are used to identify concerns. The computation results are in agreement with experimental data. The results with the ISN show that there is no obvious shock wave or expansion wave in the test section, test area is greatly expanded, and the flow quality is also quite uniform. The aerodynamic forces and moments of the AGARD-B model in the whole test section agree well with the reference data.

The ISN expands the scope of the test area,and the size and attitude angle of the test model can be enlarged appropriately.The technique can be used for future wind tunnel nozzle designs.

Acknowledgment

This study is supported by Supersonic Laboratory of CAAA.and National Nature Science Foundation of China(Nos.11672283, 11872349).