Design of a continuously variable Mach-number nozzle

Transcription

1 DOI: /s Design of a continuously variable Mach-number nozzle GUO Shan-guang( 郭善广 ), WANG Zhen-guo( 王振国 ), ZHAO Yu-xin( 赵玉新 ) Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha , China Central South University Press and Springer-Verlag Berlin Heidelberg 2015 Abstract: A design method was developed to specify the profile of the continuously variable Mach-number nozzle for the supersonic wind tunnel. The controllable contour design technique was applied to obtaining the original nozzle profile, while other Machnumbers were derived from the transformation of the original profile. A design scheme, covering a Mach-number range of 3.0<Ma<4.0, was shown to illustrate the present design technique. To fully validate the present design method, computational fluid dynamics (CFD) analyses were carried out to study the flow quality in the test area of the nozzle. The computed results indicate that exit uniform flow is obtained with 1.19% of the maximal Mach-number deviation at the nozzle exit. The present design method achieves the continuously variable Mach-number flow during a wind tunnel running. Key words: wind tunnel; variable Mach-number nozzle; flow quality; method of characteristics; numerical validation 1 Introduction As the aerodynamic performance of the aircraft has improved over past decades, the necessity for improvement in wind tunnel capability has increased. For a conventional wind tunnel, it is necessary to provide one nozzle for each discrete Mach-number [1]. To fully investigate the dynamic characteristics of the vehicles, there is, therefore, a current need within the supersonic community for an accurate, continuously variable Mach-number nozzle [2]. The purpose of a continuously variable Mach-number nozzle for the wind tunnel is to provide conditions in the test region having a range of desired continuous Mach number flow during a wind tunnel running. There are many techniques that have been used to provide a variable nozzle, such as the plug-type [3 5] and flexible nozzles [6 8]. The resultant unwanted wakes and shock waves that emanate from the plug nozzles result in flow quality that is much worse. Therefore, the plug nozzle is only applied as the exhaust nozzle of the rocket in the present. The most versatile method of obtaining variable nozzle for the wind tunnel is to utilize the flexible wall with multiple jacks or a single jack to control its shape. A simple type of flexible nozzle was first proposed in 1955 by ROSEN [6]. The flexible contour design has been widely applied to the two-dimensional nozzles. VARNER et al [1] summarized the design techniques of the flexible two-dimensional nozzle. JUHANY and HUSAINI [9] increased the Mach number range of an existing wind tunnel utilizing a single-jack nozzle. However, this type of nozzle requires quite complex systems for control and plate stress monitoring which is expensive to construct. Moreover, the current flexible nozzle only produces a discrete Mach-number exit flow during a wind tunnel running, that is, this nozzle could not provide the desired flow conditions for dynamic characteristic research of the supersonic vehicle. In order to satisfy the current need of the vehicle investigation, a design technique of the continuously variable Mach-number nozzle for the two-dimensional wind tunnel is developed in this work. The present design technique is divided into two sections: the original nozzle contour design and the transformation of the nozzle contour. The original nozzle contour is designed by the controllable contour technique. This design technique is based on the method of characteristics plus boundary layer correction, and an innovative technique is added in this design process. This nozzle design method does not utilize source flow assumption, an indispensible part of the conventional nozzle design [10], and improves the flow quality in the test section. Then, the original contour is transformed to the contour with the desired exit Mach number. Foundation item: Project( ) supported by the National Natural Science Foundation of China Received date: ; Accepted date: Corresponding author: ZHAO Yu-xin, Associate Professor, PhD; Tel: ; zyx_nudt@163.com

2 523 2 Design method 2.1 Basic design method This section primarily discusses basic design method of the nozzle, the method of characteristics plus boundary layer correction. First, the method of characteristic (MOC) is utilized to determine a suitable inviscid contour and then the reference temperature solution method to the momentum integral relations is used to compute the turbulent boundary layer displacement thickness [11]. To each ordinate of the inviscid contour, the boundary-layer correction is added to obtain the physical contour of the nozzle. The method of characteristics works on the following three-step process [12 13]. First, the paths in space along which the flowfield derivatives are indeterminate are defined. These are known as the characteristics shown in Fig. 1. Second, the continuity, momentum, and energy equations are defined along the characteristics. These equations are called the compatibility equations. The third step is to solve the compatibility equations point by point on the characteristics. This is known as unit processes. The combination of the characteristic equations, the compatibility equations and the unit processes methods is what makes the method of characteristics a powerful tool for the design and analysis of the nozzle contour. Once the inviscid contour is determined, a correction is made to account for the displacement thickness of the boundary layer. The underlying assumption behind this method is that the boundary layer thickness is small compared with the nozzle radius, allowing the nozzle flowfield to be treated as inviscid for designing the characteristics [14]. There have been considerably more empirical methods to predict the boundary-layer growth [15 17]. Many correlations have been attempted to transform compressible data, obtained under conditions of heat transfer and longitudinal pressure gradients, to an equivalent incompressible case [18]. In the present contour design technique, the reference temperature method is utilized to correct the boundary-layer thickness which is an approximate engineering method for predicting skin fraction and heat transfer for the compressible flow. For a two-dimensional viscous flow, the momentum integral correlation [14] is 2 M H M Cf sec x 1 2 x d 2 d d d 2 M 1 M 2 (3) where f is the momentum thickness, θ is the flow angle, H is the shape factor of the boundary layer, γ is the specific heat ratio, M is Mach number, and C f is skin friction coefficient. Equation (3) is an ordinary differential equation, so the displacement thickness of the boundary layer can be computed with the four-stage Runge Kutta method. 2.2 Variable Mach-number nozzle design The design of the continuously variable Machnumber nozzle is divided into two sections: the original contour design and the contour transformation. A sketch showing this nozzle design concept is shown in Fig. 2. The original nozzle profile is generated by the controllable contour design technique where the Mach- Fig. 1 Illustrating schematic of characteristics dy dx The irrotational characteristic equation is tan The compatibility equation is 1 dv sin tansin 1 dy tan V d sin yd (1) (2) where x is the horizontal ordinate, y is the vertical ordinate, θ is the local flow angle, α is the Mach angle, and V is the velocity magnitude. For the planar flow, δ=0, and for axisymmetric flow, δ=1. Fig. 2 Original nozzle contour (a) and transformation of nozzle contour (b)

3 524 number distribution along the centerline is given in advance. Then, one point is selected on the divergent contour of the nozzle, namely shared point. The profile upstream the shared point, including the convergent contour and a portion of divergent section, is shifted and rotated to obtain different uniform exit flows. The controllable contour design technique for the nozzle, shown in Fig. 2(a), is detailed as follows. 1) Both the height of the throat and flow rate of the nozzle are determined on the geometric-structure requirements of the original nozzle. 2) The transonic region is solved by the Sauer s method [19]. According to the Sauer s method, initialvalue line OT and initial characteristic OT can be solved with the radius and curvature of the throat. 3) The cubic b-spline curve is applied to describing the Mach-number distribution along the centerline T C. To get the ideal contour of the supersonic region, the method of characteristics is utilized to solve the initial expansion region OT AB and the wave-cancellation region ABCD. Computational boundary conditions for method of characteristics are initial characteristic OT and centerline T C. The point B is the inflection point of the nozzle profile. 4) The convergent profile of the nozzle can be prescribed by the bi-cubic, b-spline or Witoszynski curves. The reference temperature method to momentum integral relations is applied to solving the displacement thickness of the boundary layer to correct the contour of the nozzle. After the nozzle profile was determined, the point P is needed to be selected as the shared point. The contour upstream the shared point, EOP, is movable and the downstream, PBD, is fixed, shown in Fig. 2(b). The movable contour, EOP, is shifted and rotated with a certain regularity to change the throat height, where the transformed contour requires the first derivative of the profile in point P is continuous. Then, the nozzle contour EOPD with Ma 1 exit flow is transformed to contour E O PD with Ma 2. The continuously variable Machnozzle contour can be derived from the different throat height by transforming movable contour, EOP. In order to avoid the interference between the fixed contour and the removable in transformation, the shared point P needs to be upstream inflection point, B. As a consequence, the nozzle exit Mach-number can continuously change from Ma 1 to Ma Numerical simulation method To evaluate the validity of the design method of the continuously variable Mach-number nozzle, a fully viscous CFD calculation was carried out for nozzle contours. The finite volume method is used in the numerical simulation, and the governing equation is Navier Stocks equation: t Wd ( F G )ds 0 (4) where W represents the conservation variable, F represents the non-viscous convective flux, and G represents the viscous diffusive flux. The convective terms are evaluated by the second-order AUSM scheme, the viscous terms are handled in a central-differencing manner, and the eddy viscosity is modeled by the k ω SST turbulent model. Gas state equation is supplemented to close the equations. As the nozzle is two-dimensional, a single-block 2- D structured mesh was employed. Figure 3 demonstrates the typical grid configuration of the contraction section applied in the present study. The nozzle has been assigned grid points. To satisfy the viscosity, the local grid refinement technique is brought forward in the adjacent areas of the walls. The first cell height of y + <5. Fig. 3 Computational grid The boundary conditions are pressure-inlet at the inflow boundary, symmetry along the centerline, constant wall temperature and no slip at the wall contour, and pressure-outlet at the outflow boundary. Total pressure and total temperature for pressure inlet should be specified. 3 Analysis of flow quality In order to further examine the present design technique, the effects of both the location of the shared point and the contraction contour of the original nozzle to the flow quality of the nozzle were investigated, respectively. The contour of the Ma=4.0 nozzle is the original profile, and the x-coordinate of its throat is 0 mm. The height of the throat of the original nozzle was shifted from 8.71 to mm to obtain a range of uniform exit flow with Mach number from 4.0 to 3.0. The fully turbulent CFD simulations were conducted to analyze the flow quality of the nozzles. The computational conditions for the CFD simulation are follows: total pressure p 0 = kpa and total temperature T 0 =300 K.

4 Influence of location of shared point to flow quality For the profile of the Ma=4.0 nozzle, the inflection point is around x=150 mm. In order to avoid the interference between the movable contour and fixed profiled during the contour transformation, the location of the shared point should be in front of the inflection point. In addition, the shared point should be higher than the throat of the transformed contour; otherwise, the desired Mach-number could not be obtained. Based on the above principles, the shared points were selected at x=120, 130, 140 and 150 mm, respectively. After the movable contour is transformed, the Ma=3.0 nozzles are derived from the Ma=4.0 nozzle at different shared points. Some typical flow uniformity comparisons for Ma=3.0 nozzles are provided in Fig. 4. The exit Mach-number is plotted versus height above the nozzle centerline in Fig. 4(a), where x sp is x-coordinate of the shared point. The flow uniform of the nozzle of x sp = 130 mm is the best among four nozzles. Figure 4(b) shows the Mach-number distributions along the centerline for different shared points. It can be seen that the uniform-flow core of four nozzles is about the right size. However, the further the shared point is away from the throat, the more quickly the initial section of the nozzle expands. Table 1 shows a flow field uniformity with 80% of the nozzle exit width for Ma=3.0 nozzles at different shared points. For the detailed accuracy, each flow is presented as percent deviation from the average Machnumber of the profile extremes (the maximum and minimum values). In the table, x sp, Ma, Ma and θ max are the shared point, the exit average Mach-number, the maximum Mach-number deviation, and the maximum flow angle deviation, respectively. An acceptable deviation level is observed in the table for four nozzles by direct comparison. The minimal Mach-number deviation is ±0.64% and the minimal flow angle deviation is ±0.052, which are obtained for the nozzle with x sp =130 mm. The CFD results indicate that there is an optimum shared point for the original nozzle profile to obtain the best flow quality in the test region. 3.2 Influence of contraction contour to flow quality The divergent section 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 [20]. The contraction profile plays a more important role in the flow quality for the present nozzle because a portion of convergent profile of the original nozzle is converted to the divergent contour after being Fig. 4 CFD solution for Ma=3.0 nozzles at different shared points: (a) Mach-number distributions at nozzle exit; (b) Machnumber distributions along nozzle centerline Table 1 Flow field uniformity at exit for different shared-point nozzles x sp /mm Ma ( Ma/Ma)/% θ max /( ) ± 0.71 ± ± 0.64 ± ± 0.82 ± ± 1.22 ± transformed. Therefore, it will be necessary to study the influence of the contraction profile to the flow quality in the test area of the transformed nozzles. Figure 5 shows the contraction contours selected for evaluation, which are bi-cubic, b-spline and Witoszynski curves. Each of these contours matches the radius of curvature of the supersonic contour at the throat. CFD computations are for the entire nozzle, including both convergent and divergent contours. The focus of the computation is the effect of the contraction design on the exit flow uniformity while the supersonic contour is fixed. The following cases are for Ma=3.0 nozzles, which were transformed from Ma=4.0 nozzles with different contraction profiles. Figure 6 compares the Mach-

5 526 Fig. 5 Different contraction contours reflected several times. Combining Figs. 5 and 6, it can be concluded that the bigger radius of curvature of the contraction profile near the throat could benefit the flow quality in the test section. Table 2 shows a summary of the computational results. An acceptable level is also indicated in the table by direct comparison. For the three nozzles, the maximum value of the Ma/Ma is ±1.96%. The scheme of the bi-cubic contraction profile is dramatically better than the others. The above results illustrate that the nozzle with bi-cubic contraction would therefore be the most suitable configuration among three convergent section designs. In addition, it is concluded from the CFD analysis that the contraction section of the nozzle has a significant effect on the flow uniformity at the nozzle exit. Table 2 Flow field uniformity at exit for nozzles with different contraction contours Contraction contour Ma ( Ma/Ma)/% θ max /( ) bi-cubic ± 0.64 ±0.052 b-spline ± 1.42 ±0.086 Witoszynski ± 1.96 ± Final design Based on the above analysis, the shared point of x=130 mm and bi-cubic curve of contraction section for the Ma=4.0 nozzle were selected to obtain Ma= contours by transforming the movable section. Figure 7 demonstrates the selected five nozzle profiles for CFD computations, the throat height and horizontal position of which are specified in Table 3. Mach-number contours inside nozzles are shown in Figure 8. It can be seen that Mach-number contours vary slightly and the sufficiently wide uniform-flow core exists. No compression waves or shock with significant magnitude can be observed inside the nozzles. However, small non-uniformities due to the expanding flow are Fig. 6 CFD solution for nozzles with different contraction contours: (a) Mach-number distributions at nozzle exit; (b) Mach-number distributions along nozzle centerline number distributions at the exit and along the centerline for nozzles with different contraction contours. As can be seen from Fig. 6(a), exit flow uniformity of the nozzle with the bi-cubic contraction contour is the best among three curves, while three cases have generally acceptable Mach-number uniformity. The Mach-number profile along the centerline for the nozzle with the Witoszynski contraction contour has more than two inflection points, which illustrates that compression waves or shocks, generating from the initial-expansion section, has been Fig. 7 Contours of final design (only showing convergent and portion of divergent contour)

6 527 Table 3 Flow field uniformity at exit of different Mach number nozzles in final design Ma ( Ma/Ma)/% θ max /( ) x th /mm h th /mm ± 0.07 ± ± 0.57 ± ± 0.87 ± ± 1.19 ± ± 0.64 ± Fig. 9 CFD solution for nozzles of final design: (a) Machnumber distributions at nozzle exit; (b) Mach-number distributions along nozzle centerline 4 Conclusions Fig. 8 Final design results: (a) Ma=4.0; (b) Ma=3.73; (c) Ma=3.5; (d) Ma=3.25; (e) Ma=3.0 observed at the low-mach number nozzle exit. The Mach number distributions at the exit and along the centerline are shown in Fig. 9, covering a Machnumber range of 3.0<Ma<4.0. The Mach-number distribution at the nozzle exit shows the existence of enough-wide uniform core flow. The Mach-number deviation at the exit of Ma=3.0 and Ma=4.0 nozzles are relatively small, whereas the one of the median Machnumber contour is relatively large. A detailed summary of CFD results is listed in Table 3. The Mach-number deviation from uniformity is less than 1.19% and the greatest flow angle deviation is The Mach-number deviation of the Ma=4.0 nozzle is the smallest because it is the original design point, which is designed by aerodynamically rigorous method of characteristics. The CFD results indicate that the final design nozzles provide the acceptable flow quality with excellent engineering accuracy in the test area. 1) A straightforward design technique has been developed to quickly determine the contour of the continuously variable Mach-number nozzle for the two-dimensional supersonic wind tunnel. This nozzle can yield a range of continuous Mach-number flow in the test area during a wind tunnel running. 2) The effects of both the contraction profile and the shared point on the flow quality are investigated. The primary findings are that the contraction profile significantly affects flow uniformity in test section, and there is an optimum shared point to produce the most uniform flow in the test section. 3) In the final design studies, CFD analysis indicates that flow quality is acceptable with the maximum Ma/Ma=1.19% and the nozzle does deliver the required uniform flow. References [1] VARNER M O, SUMMERS W E, DAVIS M W. A review of two-dimensional nozzle design techniques [R]. Reston: American Institute of Aeronautics and Astronautics, [2] FETTERJPFF T P, BIRFOTT J W. Overview of the advanced

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