FP04-2012-wu.pdf
TANDEM NOZZLE SUPERSONIC WIND TUNNEL DESIGN (1)
(2)
Jie Wu , Rolf Radespiel (1)
Technische Universität Braunschweig, Institute for Fluid Mechanics, Bienroder Weg 3, 38106 Braunschweig, Germany,
[email protected] (2) Technische Universität Braunschweig, Institute for Fluid Mechanics, Bienroder Weg 3, 38106 Braunschweig, Germany,
[email protected]
ABSTRACT A new tandem nozzle supersonic wind tunnel was designed for Technische Universität Braunschweig. Based on the infrastructure of the existing M=6 Ludwieg Tube in Braunschweig (HLB), two nozzles in tandem configuration are designed to get supersonic flow with a similar test section size. The first nozzle and the intermediate settling chamber serve as a throttling device to provide the flow correct mass flow for operating the second nozzle, which expands into the M=3 test section. Preliminary design trades are presented on the basis of one-dimensional flow analysis. Using alternate methods, i.e., the methods of characteristics, Witozinsky curve of contraction, as well as the Hall method for the throat design, contours of the second nozzle are designed. Further steps of numerical optimization of the second nozzle are accomplished using the DLR TAU-Code, by which reliable estimates of flow uniformity in the test section are obtained. In order to achieve a uniform flow in the test section, suited flow straighteners are employed in the settling chamber and analyzed by both, compressible engineering flow theory and RANS solutions. The flow analysis determines a reasonable configuration of the overall tandem nozzle wind tunnel and yields realistic estimates of flow uniformity in the test section. In conclusion the present work provides quantitative design trades, detailed flow quality and performance data for cost-efficient extensions of hypersonic Ludwieg tubes into the supersonic flow range. 1 INTRODUCTION Motivated by need for supersonic flow studies, an axisymmetric M=3 tandem nozzle supersonic wind tunnel is designed at Technische Universität Braunschweig. Considering power and cost requirements, the intermittent type supersonic wind tunnel is chosen. An attractive design scheme is the Ludwieg tube, which was first conceived by H. Ludwieg in 1955 [1]. The Ludwieg tube is employed in supersonic and hypersonic wind tunnels because of its low cost and high Reynolds number, as well as its reasonably long running time. It consists of four sections: high pressure gas storage tube, a nozzle, a test section and a vacuum dump tank [2]. Designing a M=3 supersonic Ludwieg tube from scratch is not a difficult task, but the construction cost is significant. In Braunschweig the Hypersonic Ludwieg Tube (HLB) exists, and hence storage tube, control valves, test section and dump tank may be reused along with the wind tunnel infrastructure. This offers significant cost savings. HLB is a M=6 hypersonic flow facility, its unit Reynolds number is up to 30 million and the test section size is 500 mm [3]. In such a case, the size of test section will be limited by the smallest cross section in the valve for a given test section Mach number according to Schrijer [4]. Simply increasing the throat area
cannot work since the flow will not reach sonic condition at the throat. A possible way to solve such problem is to add another nozzle and a settling chamber, which serve together as a throttling device to yield a correct mass flow for the second nozzle with the large throat. Thus the tandem nozzle supersonic wind tunnel is introduced. This tunnel type is not widely used at present, but it is a cost-efficient way to design a supersonic wind tunnel on the basis of an existing hypersonic wind tunnel. A typical configuration of tandem nozzle supersonic wind tunnel is represented in Fig.1.
Figure 1. Sketch of tandem nozzle wind tunnel The present work deals with the M=3 tandem nozzle supersonic wind tunnel design based on HLB facility data. First, the theoretical design is carried out by employing various methods. The second nozzle is carefully designed together with Witozinsky curve for the subsonic contraction,
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Hall’s method to design the throat region, methods of characteristics (MOC) to design the supersonic section and Rotta’s boundary layer correction method. Then, the straightening device settings in the settling chamber are analysed with compressible flow theory. The contour of the second nozzle is analyzed by numerical flow simulations based on the RANS equations. RANS simulations are also used to determine a suited number of flow straighteners and the opening angle of the first nozzle. Finally, the whole tunnel flow is numerically simulated and the results are carefully analyzed.
Subscript 0 denotes the initial state while 1 characterizes the condition in the storage tube after the valve is operated. Subscript t denotes the stagnation condition, Τ is temperature, Ρ is the pressure and Μ is Mach number, γ is the ratio of specific heats for a perfect gas. M1 is determined by the diameter ratio of the storage tube to the first throat area [5], as Eq.3. d1 d 1_ st
Pt ,1_ st Pt ,2 _ nd
T0
γ −1 2 M1 2 γ −1 (1 + M1 )2 2 γ
γ − 1 2 γ −1 Pt ,1 1 + 2 M1 = P0 (1 + γ − 1M1 ) 2 2
2
(4)
2.2 The first nozzle design The first nozzle accelerates the flow to supersonic first. Then the flow turns into a mixed flow containing subsonic and supersonic flow regions because of the interaction of backpressure and viscous boundary layer, which means there will be a system of shocks and flow separations in the first nozzle. In view of this feature, the detailed contour of the first nozzle is less important while the opening angle governs mainly the length of the first nozzle. Therefore, a conical nozzle design with proper opening angle is employed here. The total pressure drop between the first throat and the second throat is generated by shock waves and viscous losses in the settling chamber. We note that if a large part of the pressure drop is generated by shocks, the shocks will exhibit strong fluctuations in time. However, for preliminary design work a normal shock may be assumed within the the first nozzle as shown in Fig.2.
(1)
1+
D2 _ nd = D 1_ st
where D is the diameter of the cross section of throat. Subscript 1_st denotes the first throat and 2_nd refers to the second throat. With Eq.3 the total pressure of the second nozzle is known, which means the total pressure drop between the first nozzle and the second nozzle is fixed when the throat areas are given. The total pressure may be used to compute the resulting flow state in the test section for a given test section Mach number.
2.1 Storage tube design The storage tube is an important section of the whole design with high cost. A simple gas dynamic description is presented here, for more design details see [5]. Pressurized air flows into the nozzle when the valve is opened. As a result an expansion wave travels upstream into the storage tube. As the startup of continuous flow is relatively quick, this process may be regarded as impulsive [6]; therefore the expansion wave may be described by a centered expansion wave. Total flow variables change during the expansion wave according to Eq.1 and Eq.2, for details see [7].
=
(3)
where d is the diameter of cross section. Flow along the total nozzle can be considered as adiabatic flow. Assuming that sonic flow exists both in the first and second throats, the total pressure ratio between the first nozzle and the second nozzle is written in Eq.4 according to [8],
2 THEORECTICAL DESIGN The operation principle of tandem nozzle wind tunnel is as follows. High-pressure and hightemperature air is stored in storage tube. The storage tube is separated from the first supersonic nozzle by a fast-opening valve. As the valve opens, the air expands into the first nozzle with a sonic throat. The subsequent flow in the supersonic part of the first nozzle and in the settling chamber is assumed to generate significant losses of total pressure by which the mass flow density is reduced so that the flow is sonic at the large throat of the second nozzle as well. These losses are generated by shock waves and by viscous losses. Note that the detailed design of the settling chamber is important to obtain a homogeneous, steady flow in the wind tunnel test section. The design of the tunnel is carried out by dividing it into five sections: the storage tube, the first nozzle, the settling chamber, the second nozzle, and the test section.
T t ,1
γ +1
2
2( γ −1) = 1 2 1 + γ − 1 M12 M + 1 2 γ !
(2)
2
The total pressure drop coefficient correction is shown in Fig.3 as an example. The other method to calculate pressure drop is used by Emanuel [11]. It divides the flow through the straightener into 3 stages as sketched in Fig.4. Here the flow is assumed isentropic from point 1 to point 2, viscous energy dissipation takes place from point 2 to point 3 and irreversible subsonic compression is assumed between point 3 and 4.
Figure 2. Normal shock in the first nozzle Using this inviscid model the flow within the first part of the nozzle accelerates as long as it is supersonic. It turns into subsonic flow after shock, and then decelerates until the settling chamber.
Figure 4. Flow states along flow straightener
2.3 The settling chamber design The total pressure drop between the first throat and the second throat is fixed, and a significant part of it is generated by the shock in the first nozzle. However, a large pressure drop within the settling chamber will be used to obtain homogeneous flow in the test section. Settling chambers usually employ flow straighteners, such as honeycombs or porous plates for promoting uniformity of the flow and for reducing turbulence in the air stream [9]. Different from flow straighteners used in low subsonic flow, the flow straighteners used here need to withstand a remarkably large aerodynamic load, hence rather solid flow straighteners are necessary. Moreover, the flow he should be considered as compressible since significant changes in pressure and density are expected. Flow compressibility may be taken into account following two approaches. One method is recommended by Pinker and Herbert: One obtains the pressure drop coefficient with incompressible theory first. Then the ratio of pressure drop coefficient for compressible flow is obtained as an empirical correction that depends on flow Mach number and porosity [10].
While the isentropic flow acceleration from point 1 to 2 is computed from simple compressible flow theory, process from point 2 to point 3 is nonisentropic because of viscous effect. The relation between Mach number and friction coefficient is given by Eq.5 [12]: 4c f
L* 1 − M 2 γ + 1 (γ + 1) M 2 = + ln[ ] D γM 2 2γ 2 + (γ − 1) M 2
(5)
where c f is the mean friction coefficient and chosen to be 0.005 here, M is the Mach number at station 2. L* is the length of flow tube for which sonic flow conditions occur, and D is the diameter of the holes in the flow straightener. When the flow leaves the straightener, it diffuses into the station 4. Based on momentum equation, the flow condition at station 4 is obtained. According to Emanuel’s work [11], the Mach number and pressure relations can be written as Eq.6 if the inflow Mach number is reasonably small.
M 1 = βM 2 = β (
P4 P ) M 3 = ( 4 )M 4 P1 P1
(6)
where β is the porosity of flow straighteners. With the two methods above, the number of flow straighteners can be estimated so that the desired total pressure drop is obtained. 2.4 The second nozzle design The second nozzle is the core of the tandem nozzle. Its design is composed of three parts: Contraction, throat and supersonic section. The contraction accelerates the flow from subsonic to sonic at the throat of the nozzle. The throat allows a smooth transition from subsonic flow to supersonic flow. The supersonic section is designed to expand the flow to the test section design Mach number. Since the contour of the nozzle directly affects the flow quality in the test
Figure 3. Pressure drop coefficient ratio between compressible and incompressible flow [10]
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limiting characteristic. At the same time, an initial data line which is downstream of the limiting characteristic is chosen to start the method of characteristics solution (HA-AA). In the procedure of flow field calculating, the limiting characteristic serves for the initial data line. More details are given in reference [18].
section, the design of nozzle contour is important. The method of characteristics (MOC) is used frequently in supersonic nozzle design assuming that no shock and hence entropy changes occur. This approach allows rapid design computations. It is the design approach followed in this paper. There are two steps in designing a nozzle by using MOC. First one can assume the flow to be ideally inviscid, which is used to generate an initial potential flow contour. Then one can estimate the development of boundary layer parameters in order to add a boundary layer correction to the initial contour. 2.4.1 Contraction design For the contraction design, there is no specific rule which is better than any others. Several methods are available for the designing a wind tunnel contraction, such as Witozinsky curve [13], fifth-order polynomial by Bell and Mehta [14] and matched cubic curves by Morel [15]. For moderate contraction ratios the details of the geometric rules employed appear less important if the contour is sufficiently smooth to avoid flow separation. The known design variants are therefore compared in Chapter 3 by using numerical flow computations.
Figure 5. The sketch of expansion and straightening section The limiting characteristics HA-AI also provides flow direction information to HA-AA. A point along the Hall contour is specified to be the beginning point to generate the expansion contour; this contour uses gradually increasing turning angles by which the supersonic flow will expand. Computing the streamline equation, the contour of expansion section is generated until the turning angle reaches its prescribed maximum value.
2.4.2 Throat design The throat region is of considerable importance because the flow conditions just downstream of the narrowest section are required in order to start the supersonic design. There is no conclusion about the best approach to calculate the sonic line, but the general guideline is to avoid an over curved sonic line. The theory used here to calculate the flow field is due to Hall [16]. The profile near the throat is assumed to be a circular arc. The radius of curvature is the important parameter, and it depends mostly on experience. A large radius of curvature is desired, but it is impossible to keep the sonic line straight in axisymmetric flow. The commonly used radius of curvature is about 4-8 times the throat radius. However, it appears somewhat arbitrary to get a proper radius of curvature for specific nozzle design. This will be demonstrated with numerical simulations in Chapter 3.
2. Straightening section design The portion of the contour where the turning angle keeps decreasing is the straightening section, which is designed to eliminate the expansion waves generated by the expansion section. Assume XD to be an arbitrary point along CD-AD. A plane radial flow is defined, see Fig.5, for which all the variables are a function only of the distance r from a fixed point in the plane, according to McCabe [19]. Thus, the Mach number distribution along characteristic CD-AD is given as: γ +1
r 1 1 γ − 1 2 2 (γ −1) = 1+ M 2 r * M γ + 1
(7)
r* is the sonic radius of radial flow. Using the principle that the velocity change in wave is normal to the wave [8] and Eq. 7, the turning angle Θ acts as Eq. 8 and has a certain range with a maximum value of half Prandtl-Meyer angle at the design Mach number in the test section,
2.4.3 Supersonic section design 1. Expansion section design Usually, the supersonic part of the nozzle is composed of two sections: the expansion section which quickly accelerates the flow and the straightening section composed of a series of simple waves. In the designing the contour of the supersonic section with MOC, a limiting characteristic needs to be specified first according Anderson [17]. In Fig.5, HA-AI is chosen to be the
γ +1 γ −1 2 arctan (M − 1) + c (8) γ −1 γ +1 In Eq. (8), c is a constant. For the given turning θ = arctan M 2 − 1 −
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Mach number. When the incoming Mach number is large, the deflection angle of shock will be small which means a larger model size can be chosen. These considerations were computed numerically for the HLB tandem nozzle by taking into account the maximum allowed storage tube pressure of 30 bar. The design parameters, these are test section diameter, test section Mach number and total temperature were varied based on the gas dynamic equations of Chapter 2.1. The resulting Reynolds numbers for the generic capsule and rocket models are displayed in Fig. 6 and Fig. 7, respectively, along with trajectory data of Apollo and Ariane 5.
angle value at point CD in Fig. 5 it is possible to solve the Mach number, and therefore the Prandtl-Meyer angle ν is known by the PrandtlMeyer equation [8].
ν=
γ +1 γ −1 2 arctan ( M − 1) − arctan M 2 − 1 (9) γ −1 γ +1
After that, the turning angle is decreased gradually until zero is obtained, and the radius for each point along characteristic CD-AD will be obtained. Thus the flow information on the characteristic CD-AD is known. Based on continuity equation, the contour of CD-CE is computed from algebraic relations. 2.5 Boundary layer correction The nozzle contour from the design method described above is just a potential flow contour. However, a boundary layer grows along the wall of the nozzle. This is taken into account by displacing the potential flow contour away from the center axis by using the displacement thickness of the boundary layer. There are several approximate methods to predict the rate of the boundary layer growth along the nozzle wall as summarized by Thompson [20]. In the present design the boundary layer thickness from the throat to the straightening section was calculated by using an expression given by Rotta, for more details refer to [21].
Figure 6. Maximum model Reynolds number as a function of Mach number for Apollo capsule [22]
2.6 The test section design The design of the test section is to allow for supersonic flow around the wind tunnel model at a reasonable Reynolds number that is virtually free of any interference with the test section wall. The size of the wind tunnel model is restricted by two effects: The start up problem of blunt wind tunnel models and the adverse effects of reflected shock waves on slender configurations. Both problems depend on the test section Mach number and its diameter and they restrict the attainable Reynolds number of the model. In the present design we investigate these design constraints using two generic wind tunnel models: A rocket model with a length–to-diameter ratio of 4.5 that represents the Ariane 5 afterbody and a blunt capsule model that represents the Apollo reentry capsule. Ariane 5 and the Apollo capsule exhibit two different aerodynamic flow problems. Apollo capsule model is a blunt body with a strong detached shock. Recent experience shows that successful start up of the capsule flow is obtained at a ratio between model and test section diameters of 0.36. Ariane 5 model is a pointed body, where the allowed size of interference free flow in the test section varies with the incoming
Figure 7. Maximum model Reynolds number as a function of Mach number for Ariane 5 rocket [23] It appears that the obtained Reynolds number depends on flow total temperature more than on the test section size. The maximum Reynolds number of the tandem nozzle test section covers flight Reynolds numbers of the Apollo Capsule between M=3 and M=6, given the operational limitations of the HLB. Hence it is decided to choose the test section Mach number of 3 for tandem nozzle design. Note, that the flight Reynolds numbers of Ariane 5 model are significantly larger than the model Reynolds numbers for M
