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Unsteady effects in critical nozzles used for flow metering
Measurement 29 (2001) 1–10 www.elsevier.com / locate / measurement
Unsteady effects in critical nozzles used for flow metering a, a b b E. von Lavante *, A. Zachcial , B. Nath , H. Dietrich a
Institute of Turbomachinery, University of Essen, D-45127 Essen, Germany b ELSTER Produktion GmbH, D-55252 Mainz-Kastel, Germany Received 10 September 1999; accepted 28 February 2000
Abstract A class of critical nozzles, shaped according to the recommendation of the ISO Standard 9300, was selected for the present study. The flow in the nozzles was investigated numerically and experimentally, making a validation of the results by direct comparison possible. In the numerical part, the unsteady compressible viscous flow was simulated by an appropriate Navier–Stokes solver. The experimental work employed flow visualization and accurate pressure measurements to analyze the flow. Significant unsteady phenomena were discovered, leading in some cases to a decrease in the flow rate. The experimental and numerical results were in good qualitative and quantitative agreement. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Navier–Stokes; Critical nozzles; Flow metering
1. Introduction One of the simplest methods of highly accurate measurement of mass flows is the employment of critical nozzles. As a matter of fact, sonic nozzle test rigs are often used as reference flow meters [9,10,12–14]. The guidelines and standards in using these devices are well defined and covered by the ISO Standard 9300 [10]. It is valid for toroidal nozzles with critical Reynolds numbers between Re 5 10 5 and Re 5 10 7 . However, critical nozzles of significantly smaller diameter are also in frequent use [9,12,15]. In this case, relatively large fluctuations in the mass flow have been observed, making the *Corresponding author. Tel.: 149-201-812-3909; fax: 149201-812-3909. E-mail address: [email protected] (E. von Lavante).
calibration of these nozzles difficult. In particular, nozzles of diameters less than 1 mm and Reynolds numbers as small as 3 3 10 3 display higher measurement uncertainties. The aim of the present research is to investigate the flow behavior in sonic nozzle test rigs in general, and the instabilities in small sonic nozzles in particular. In the past, flows in supersonic and transonic nozzles have been studied quite extensively. Due to their significant importance to the aerospace engineering, especially in the area of rocket propulsion, they were investigated numerically and experimentally numerous times [1–3,11]. However, most of the research work concentrated on the overall features of these nozzles, assuming steady flow and leaving out some of the unsteady details. Besides, the size, and therefore the Reynolds numbers of these nozzles were significantly larger than considered in the present work. Therefore, the present study could not
0263-2241 / 01 / $ – see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S0263-2241( 00 )00021-X
2
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
take advantage of the previous results and findings. Consequently, it was decided by the present authors to carry out a numerical and experimental analysis of the metering nozzles described above. A representative critical nozzle geometry, as recommended by the ISO Standard 9300, was selected for the baseline investigation. The nozzle shape is shown in Figs. 2 and 3. It is a typical convergent– divergent nozzle, with a toroidal throat having a diameter of 10 mm, and a divergent part of 70 mm length, with a divergence wall angle of 48. The Reynolds number based on the critical conditions and the throat diameter was approximately Re 5 1.5 3 10 5 . Additionally, in order to investigate the influence of the critical Reynolds number on the flow behavior, three other Reynolds numbers were used in the numerical study: Re 5 5.0 3 10 5 , Re 5 1.5 3 10 6 and Re 5 1.0 3 10 7 . There were two versions of the nozzle produced and, consequently, also simulated. One had a two-dimensional shape, with upper walls made of solid aluminum, and side walls of glass in order to allow optical visualization; the other nozzle was manufactured out of stainless steel and had the usual annular shape. Accordingly, the simulation assumed in the first case a two-dimensional flow, and in the second case an axisymmetric geometry. The physical flow conditions are discussed below.
2. Numerical approach The governing equations to be solved in the present simulations are the two-dimensional or axisymmetric compressible Navier–Stokes equations. They are, in a simplified vector form in general, body-fitted coordinates j and h in the weak conservation law form: ≠Q ≠(F 2 Fv ) ≠(G 2 Gv ) S ] 1 ]]] 1 ]]] 5 ] ≠t ≠j ≠h J
(1)
The above equations are given in more detail by, for example, Steger [4], and will not be repeated here. Eq. (1) is integrated in time by solving its semidiscrete form by means of modified Runge–Kutta (R– K) time stepping. The presently used two stage version of the R–K procedure can be written as
Q 1 5 Q n 2 a1 Dt * RsQ nd Q 2 5 Q n 2 a2 Dt * RsQ 1d Q
n11
5Q
(2)
2
with the residual R given by ≠(F 2 Fv ) ≠(G 2 Gv ) ≠Q ] 5 2 ]]] 2 ]]] 5 R ≠t ≠j ≠h
(3)
The coefficients a1 and a2 were optimized by von Lavante et al. [5] for maximum multigrid performance (damping of high frequencies). That optimization was, however, done using a linear hyperbolic model equation. In realistic applications, these coefficients worked fine for the inviscid Euler equations as well as for some viscous cases. The simple two stage R–K procedure was as efficient as the more frequently used four stage scheme, but required only one optimized coefficient, a1 5 0.42. Due to the complexity of the predicted flow, a simple numerical scheme with central spacial differences and artificial dissipation added explicitly was not suited for the present simulations. Therefore, the well known and proven Roe’s Flux Difference Algorithm [7] was subsequently employed. The numerical scheme is based on Roe’s Flux Difference Splitting in finite volume form, as developed by El-Miligui et al. [6]. In previous numerical predictions, this scheme was highly effective in providing accurate viscous results for wide ranges of Mach numbers. The reconstruction of the cell-centered variables to the cell-interface locations was done using a monotone interpolation of the primitive variables vector Q p 5s r, u, v, pd T . The interpolation slope was limited by an appropriate limiter, according to the previously published MUSCL type procedure (see, for example, [6]). The viscous fluxes Fv and Gv were centrally differenced. In the present work, the boundary conditions were implemented with the help of dummy (ghost) cells. At the subsonic inflow boundary, a locally onedimensional weakly non-reflective boundary condition, based on the isentropic theory of the Riemann problem, was used. The tangential velocity, incoming Riemann invariant, total pressure and entropy in the first ghost cell were specified. The outgoing Riemann invariant was extrapolated from the interior domain. At the subsonic outflow boundary, the static
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
pressure was specified, subject to the fluctuations allowed by the non-reflective treatment. The remaining variables were extrapolated. At solid walls, the velocities were anti-reflected, resulting in zero velocity on the boundary. The static pressure and density were reflected, resulting in zero gradients of these variables at the wall (adiabatic wall). Finally, at the interzonal boundaries, the block grids were overlapping by two cells, providing smooth transitions of the dependant variables Q due to the present MUSCL extrapolation in the projection phase.
3. Experimental approach
3.1. Laser light sheet procedure The laser light sheet procedure is a well-known method to visualize flow patterns in flow fields in general and in gas measuring and controlling units in particular. For that purpose, a light sheet is generated that illuminates a flow field to be examined in a thin plane layer (thickness: approx. 0.5 mm). Within this layer, the flow behavior can be observed using the tracer streaks of scattering particles. Fig. 1 shows the test setup of the above described procedure to record flow images by means of the laser light sheet. The flow in the test nozzle, equipped with optical inputs, is optically accessible for a camera through an inspection window. The laser light sheet is generated by the optical inputs on the top and bottom of the test nozzle. A vacuum blower generates the flow through the
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test setup. The extent of the negative pressure downstream of the nozzle is adjusted by means of a bypass valve which is installed between the test nozzle and the blower. To visualize the air flow, an aerosol generator was used to add DEHS (diethyl sebacate) particles in sizes between 0.2 and 1.0 mm. When these particles suspended in the flow pass the light sheet, they scatter the laser light and are therefore visible. Within the light sheet, an image of the flow pattern appears that is recorded by a video camera through the inspection window. The analog output signal of the camera is digitized by a frame grabber board installed in a PC and then stored. Eventually, a digital post-processing of the images will be carried out. In this case, a nozzle-shaped rectangular channel is used. The light sheet is arranged in such a manner that the camera is able to record the flow along the axis of the nozzle.
3.2. Pressure measurements of nozzle flow In order to measure the static pressure of the flow along the nozzle wall, 18 pressure measuring locations, shown in Fig. 5, were distributed between the throat and the exit end of a nozzle with a diameter of the throat d th 510 mm. The individual pressure measuring points have been arranged parallel to the direction of the flow and alternate in two series set off by 908 (see Fig. 2). Each pressure measuring tap was threaded to allow the installation of a pressure sensor. The pressure was measured by a piezoresistive miniature precision pressure transducer. The sensor had an outside diameter of d52 mm and was integrated in a hollow bolt. It could therefore be inserted into any of the individual measuring points in a reproducible way. The openings not in use were hermetically plugged by proper Allen screws.
4. Results
Fig. 1. Schematic diagram of the test setup.
Initially, the exact flow character at the conditions given by the ISO 9300 Norm was not known. It was, therefore, decided to simulate the corresponding case numerically, hoping to obtain a better idea about the flow field. As the simulation results revealed surprisingly complex features of the flow, the simulation
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E. von Lavante et al. / Measurement 29 (2001) 1 – 10
Fig. 2. Design drawing of the nozzle with the location of taps for pressure sensors.
was followed up by the experimental effort in order to validate the numerical data. First, the flow in the baseline nozzle shape, given in Fig. 3 including the computational grid, was computed. In this case, the fluid (air) was entering the nozzle at atmospheric stagnation conditions through the left boundary, with the total temperature T 0 5 293.14 K and total pressure P0 5 1013.25 mbar. The pressure difference across the nozzle was Dp 5 300 mbar. The outflow boundary is to the right, whereas the upper boundary is solid (wall) and the lower boundary represents the r50 symmetry line. The critical Reynolds number was in this case Re * 5 1.65 3 10 5 , which is at the lower limit given by the ISO 9300 Standard and corresponds to a Re ISO 5 1.31 3 10 5 . According to the real physics of the flow, the nozzle started from the zero flow condition (quiescent), the velocity being induced by the low pressure at the end of the nozzle. According to the simple quasi-one-dimensional theory, it was from the beginning obvious that the above pressure difference was too small to result in a smooth supersonic flow in the
entire nozzle. A system of complex shock waves causing a periodic flow separation at the wall developed. The separated bubbles detached from the wall and formed vortices that were convected downstream. The separated boundary layer had a noticeable blockage effect. The resulting pressure and density contours are shown in Fig. 4. The unsteady character of the flow is, of course, detrimental to the accuracy of the nozzle as a flow metering device. At this point, the authors decided to pursue two parallel investigations. First of all, an experiment had to be conceived to validate the above numerical results. Secondly, the numerical work was continued, trying to find a simple fix to the unsteadiness problem. Fig. 5 shows a diagram of the static pressure ps at four different locations in the nozzle as a function of the back pressure (pressure after the exit from the nozzle) pne . The exit pressures pne have been indicated as differential pressures with respect to the atmospheric pressure. The four locations of the static pressure measurement shown in Fig. 5 were chosen to be in front of the nozzle throat (positions 3 and 4), in the nozzle
Fig. 3. Computational grid for the baseline nozzle.
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
5
Fig. 4. (a) Simulation results showing velocity contours; (b) simulation results showing static pressure; (c) simulation results showing density contours.
Fig. 5. Pressure measured at four locations in the nozzle. The locations are indicated in the small insert.
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E. von Lavante et al. / Measurement 29 (2001) 1 – 10
throat itself (position 5) as well as in the outlet of the nozzle (position 6). As could be expected, the pressure in the nozzle throat asymptotically approaches an absolute static pressure of approx. ps 5 542 mbar at an atmospheric equilibrium pressure of patm 5995.5 mbar. This corresponds to a pressure ratio of ps /patm 50.544. This ratio is somewhat larger than the theoretical critical pressure ratio of ps /patm 50.528 given in the literature [8]. The reason for this discrepancy can be explained by noticing that the theory assumes one-dimensional, inviscid flow. As the results of the simulation clearly show, for example, in Fig. 4, the flow is of highly complex quasi-three-dimensional nature, with domains showing massive flow separation. The critical condition in the nozzle throat cannot be directly recognized as a straight isotach. Instead, the isotach is curved and located directly after the nozzle throat. At back pressures of pne 5125.5 mbar or higher, the static pressure ps at positions 3, 4 and 5 remains constant (see Fig. 5). However, the values of the absolute static pressure in front of the throat (positions 3 and 4) are at a higher level than the pressure in the
nozzle throat. According to the simple isentropic theory, they are supposed to behave this way due to their large cross sectional areas. The graph of the static pressure at a location downstream of the nozzle throat (position 6) is different, however. In the case of fairly high back pressures, the flow through the nozzle is subcritical, and the static pressure at position 6 will be higher than that in the nozzle throat because of the larger cross section. As the back pressure is further decreased, the critical conditions will be eventually reached in the throat, and therefore the flow will be supersonic at position 6, showing a static pressure lower than the critical pressure. Caution should be exercised, however; according to the numerical results, and confirmed by the experiments, purely supersonic conditions can not be maintained throughout the nozzle due to the relatively large back pressure. At a short distance downstream of the throat, a system of oblique shocks will form, causing an increase in the pressure. The supersonic flow will prevail only over short distances, for short periods of time (see below). Fig. 6 shows a comparison of the computed and
Fig. 6. Pressure along the wall of the nozzle wall.
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
measured absolute static pressures along the wall of the nozzle for Reynolds number Re * 5 1.5 3 10 5 . The outflow pressure ratio was in this case varied between pout /p0 5 0.95 and pout /p0 5 0.643. As pointed out above, at this Reynolds number, these outflow pressure ratios are too high to give smooth flow in the nozzle throat, resulting in unsteady flow with periodic unchoking of the nozzle. Contrary to the simple quasi-one-dimensional theory, the nozzle is not choked at the highest pressure ratio, indicating significant viscous effects in the diffusor. At all the higher pressure ratios, the nozzle is obviously choked. Considering the significant pressure fluctuations especially in the throat region, and the fact that in the experimental work, the time-wise averaging is different from the one used in the numerical analysis, the agreement is rather good. Principally, all graphs show similar characteristics. The flow is accelerated in the inlet of the nozzle, resulting in a reduction of the static pressure. The divergent part of the nozzle actually serves as a diffusor to accomplish the pressure recovery necessary for the relatively high exit pressure. However, it must be distinguished between three different cases. Case I is characterized by fully subsonic flow ( pne 5 50 mbar); the flow rate per area reaches its maximum, and the pressure its minimum, at the narrowest part of the nozzle, i.e. the nozzle throat (x516 mm). The diffuser reduces the Mach number, thus increasing the static pressure ps in the flow. Case II is reached for an exit pressure of approx. pne 5 130 mbar. This corresponds to a static pressure in the throat of ps 5 530 mbar. The flow will just become critical, and then follow the subcritical path to the exit. The pressure will subsequently recover the exit pressure of pne 5 130 mbar. Case III is finally obtained by further reducing the exit back pressure to pne 5 300 mbar. The velocity in the nozzle throat again reaches the speed of sound, with the corresponding critical pressure. Additionally, there is also supersonic flow in the beginning of the diverging part of the nozzle, indicated by the continuing
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decrease of the static pressure (x521 mm). This is accompanied by a corresponding increase of the flow Mach number, reaching values greater than the speed of sound. As shown in Figs. 4 and 11, the level of the exit pressure in Case III leads to the formation and shedding of vortices close to the wall. This is caused by a series of compression waves that lead to strong fluctuation of the static pressure at a given location on the wall. This effect is displayed by the rise of the static pressure ps after reaching its minimum at x521 mm. Any further decrease of the exit pressure leads to a shift of the minimum of the static pressure downstream. (x525.5 mm). An obvious method to achieve a smoother flow in the nozzle was to lower the backpressure. This, however, was deemed undesirable due to the significant increase in effort in generating it. Lastly, in order to validate the computations in the basic ‘short’ nozzle, the nozzle geometry was extended to include also a part of the duct into which the gas from the nozzle was flowing. In particular, it was necessary to prove that the flow instabilities were not caused by the outflow boundary condition. The geometry of this configuration can be seen in Fig. 7. Noteworthy is the long extension of the duct, eliminating any possibility of the outflow boundary influencing the flow in the nozzle itself. Fortunately for the investigators, the simulations in this geometry confirmed the previous findings, giving very similar results. The baseline geometry was also computed as a two-dimensional configuration (as opposed to the axisymmetric approach used in the work presented above) in order to allow a qualitative comparison with the light sheet flow visualization described above. Fig. 8 displays a detailed view of the boundary layer at the upper wall obtained in the present simulation at an axial location of approximately 38 mm after the throat. Compared to the picture obtained by the laser light sheet method (Fig. 9) at the same axial location, the fair level of qualitative agreement is apparent. The generation (shedding) of
Fig. 7. Computational grid for the extended nozzle.
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E. von Lavante et al. / Measurement 29 (2001) 1 – 10
Fig. 8. Detailed view of the upper boundary layer.
Fig. 9. Laser light sheet picture of the 2-D nozzle.
vortices in the boundary layer due to the shockboundary layer interaction can be also seen in Fig. 10. Here, the timewise development of the flow, as simulated in the present work, is presented in terms of the velocity magnitude. The nozzles in the present work are to be used for gas metering, so that the mass flow through the throat is of particular interest. Theoretically, in a choked nozzle, it should be constant with time. However, due to the pressure waves travelling upstream, the throat in the smallest nozzle (Re * 5 1.0 3 10 5 ) is periodically unchocking, resulting in an intermittent decrease of the mass flow. The computed timewise history of the mass flow, non-dimensionalized by the theoretical (inviscid, quasione-dimensional, ideal nozzle) mass flow, is shown in Fig. 11a–f. It should be noted that this expression corresponds to an instantaneous value of the nozzle
discharge coefficient CD . The mass flow histories are displayed for the original ‘short’ configuration in Fig. 3 (Fig. 11a, c and e) and the extended ‘long’ configuration in Fig. 7 (Fig. 11b, d and f) for three different Reynolds numbers. Obviously, the two nozzle configurations give very similar results. Interestingly, the higher Reynolds numbers with the corresponding decrease in the boundary layer thickness result in more stable flow conditions. At the Reynolds number Re * 5 5.0 3 10 5 , the frequency and amplitude of the unsteady pressure waves is smaller, as indicated by the smaller mass flow fluctuations in Fig. 11c and d. At Reynolds numbers Re * 5 1.5 3 10 6 and higher, the flow in the throat remains choked at all times, as evidenced by the constant mass flow in Fig. 11e and f.
5. Conclusion In the present experimental and numerical study of the flow in critical nozzles used for gas metering, good agreement between the numerical and experimental results was achieved. Both approaches indicate that the flow in these devices is highly complex, with massively separated regions and strongly unsteady pressure waves. The pressure waves will propagate upstream and cause a periodic fluctuation of the mass flow in the throat and at the exit of the nozzles. For short periods of time the flow becomes subcritical with the corresponding decrease of mass flow. The unsteady behavior, shown in Fig. 11 as a plot of the mass flow at the nozzle throat as a
Fig. 10. (a) Computational results for the 2-D nozzle, displayed as velocity magnitude contours; (b) computational results for the 2-D nozzle, displayed as velocity magnitude contours; (c) computational results for the 2-D nozzle, displayed as velocity magnitude contours; (d) computational results for the 2-D nozzle, displayed as velocity magnitude contours.
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
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Fig. 11. (a) Dimensionless mass flow as a function of time, ‘short’ nozzle; (b) dimensionless mass flow as a function of time, ‘long’ nozzle; (c) dimensionless mass flow as a function of time, ‘short’ nozzle; (d) dimensionless mass flow as a function of time, ‘long’ nozzle; (e) dimensionless mass flow as a function of time, ‘short’ nozzle; (f) dimensionless mass flow as a function of time, ‘long’ nozzle.
function of time, measured in seconds, is the result of the poorly matched back pressure to the shape of the nozzle. However, a purely supersonic flow in the nozzle would require an unrealistically low exit pressures. A compromise is needed in order to solve this problem. A major shortcoming of the present
numerical simulations is its two-dimensional character. Although no positive proof is available, the authors are certain that the real separation regions and other disturbances are not uniformly distributed around the circumference of the nozzles. The computed discharge coefficients are therefore underesti-
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E. von Lavante et al. / Measurement 29 (2001) 1 – 10
mated and should be only used to indicate certain tendencies. A genuinely three-dimensional simulation is costly but unavoidable. Clearly, more work in this research area is required if recommendations regarding the improvement of accuracy of the critical nozzles are to be made.
[7]
[8] [9]
References ¨ [1] E. von Lavante, J. Gronner, Semi-implicit Schemes for Solving the Navier–Stokes Equations, 9th GAMM Conference, Lausanne, Notes on Numerical Fluid Mechanics, Vieweg, 1992. [2] E. von Lavante, A. El-Milligui, H. Warda, Simulation of Unsteady Flows Using Personal Computers, The 3rd International Congress on Fluid Mechanics, Cairo, 1990. [3] R.J.G. Norton, Prediction of the Installed Performance of 2-dimensional Exhaust Nozzles, AIAA Paper 87-0245. [4] J.L. Steger, Implicit, Finite-Difference Simulation of Flow about Arbitrary Geometries, AIAA Journal 16 (7) (1978) 679–686. [5] E. von Lavante, A. El-Miligui, F.E. Cannizzaro, H.A. Warda, Simple Explicit Upwind Schemes for Solving Compressible Flows, Proceedings of the 8th GAMM Conference on Numerical Methods in Fluid Mechanics, Delft, 1990. [6] A. El-Miligui, F. Cannizzaro, N.D. Melson, E. von Lavante, A Three-Dimensional Multigrid Multiblock Multistage Time Stepping Scheme for the Navier-Stokes Equations, Proceed-
[10]
[11] [12]
[13]
[14]
[15]
ings of the 9th GAMM Conference on Numerical Methods in Fluid Mechanics, Lausanne, 1991. P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Comp. Physics 43 (1981) 357–372. A.H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, John Wiley & Sons, 1953. ´ B. Nath, H. Dietrich, Compteur etalon automatise´ a` tubes de ´ Venturi fonctionnant en zone critique pour l’etalonnage de ´ ` ´ ` debitmetres ou de compteurs volumetrique, 8e Congres ´ ´ International de Metrologie Metrologie 97, 20.10.–23.10.97, Besanc¸on, France. International Standard ISO 9300, Measurement of Gas Flow by Means of Critical Flow Venturi Nozzles, 1st Edition, Beuth, Berlin, 1990. J.D. Anderson, Modern Compressible Flow, Mc Graw-Hill, New York, 1990. M. Ishibashi, M. Takamoto, Discharge Coefficient of Superaccurate Critical Nozzle at Pressurized Condition, Proccedings of the 4th International Symposium on Fluid Flow Measurement, Denver, USA, June 1999. H. Dietrich, B. Nath, E. von Lavante, High Accuracy Test Rig for Gas Flows from 0.01 m 3 / h up to 25 000 m 3 / h, Proceedings of the IMEKO-XV Congress, Osaka, Japan, June 1999. M. Ishibashi, M. Takamoto, Very Accurate Analytical Calculation of the Discharge Coefficient of Critical Venturi Nozzles with Laminar Boundary Layer, Proceedings of the FLUCOME97 Conference in Hayama, Japan, Sept. 1997. G. Wendt, Private Communication.
Unsteady effects in critical nozzles used for flow metering a, a b b E. von Lavante *, A. Zachcial , B. Nath , H. Dietrich a
Institute of Turbomachinery, University of Essen, D-45127 Essen, Germany b ELSTER Produktion GmbH, D-55252 Mainz-Kastel, Germany Received 10 September 1999; accepted 28 February 2000
Abstract A class of critical nozzles, shaped according to the recommendation of the ISO Standard 9300, was selected for the present study. The flow in the nozzles was investigated numerically and experimentally, making a validation of the results by direct comparison possible. In the numerical part, the unsteady compressible viscous flow was simulated by an appropriate Navier–Stokes solver. The experimental work employed flow visualization and accurate pressure measurements to analyze the flow. Significant unsteady phenomena were discovered, leading in some cases to a decrease in the flow rate. The experimental and numerical results were in good qualitative and quantitative agreement. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Navier–Stokes; Critical nozzles; Flow metering
1. Introduction One of the simplest methods of highly accurate measurement of mass flows is the employment of critical nozzles. As a matter of fact, sonic nozzle test rigs are often used as reference flow meters [9,10,12–14]. The guidelines and standards in using these devices are well defined and covered by the ISO Standard 9300 [10]. It is valid for toroidal nozzles with critical Reynolds numbers between Re 5 10 5 and Re 5 10 7 . However, critical nozzles of significantly smaller diameter are also in frequent use [9,12,15]. In this case, relatively large fluctuations in the mass flow have been observed, making the *Corresponding author. Tel.: 149-201-812-3909; fax: 149201-812-3909. E-mail address: [email protected] (E. von Lavante).
calibration of these nozzles difficult. In particular, nozzles of diameters less than 1 mm and Reynolds numbers as small as 3 3 10 3 display higher measurement uncertainties. The aim of the present research is to investigate the flow behavior in sonic nozzle test rigs in general, and the instabilities in small sonic nozzles in particular. In the past, flows in supersonic and transonic nozzles have been studied quite extensively. Due to their significant importance to the aerospace engineering, especially in the area of rocket propulsion, they were investigated numerically and experimentally numerous times [1–3,11]. However, most of the research work concentrated on the overall features of these nozzles, assuming steady flow and leaving out some of the unsteady details. Besides, the size, and therefore the Reynolds numbers of these nozzles were significantly larger than considered in the present work. Therefore, the present study could not
0263-2241 / 01 / $ – see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S0263-2241( 00 )00021-X
2
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
take advantage of the previous results and findings. Consequently, it was decided by the present authors to carry out a numerical and experimental analysis of the metering nozzles described above. A representative critical nozzle geometry, as recommended by the ISO Standard 9300, was selected for the baseline investigation. The nozzle shape is shown in Figs. 2 and 3. It is a typical convergent– divergent nozzle, with a toroidal throat having a diameter of 10 mm, and a divergent part of 70 mm length, with a divergence wall angle of 48. The Reynolds number based on the critical conditions and the throat diameter was approximately Re 5 1.5 3 10 5 . Additionally, in order to investigate the influence of the critical Reynolds number on the flow behavior, three other Reynolds numbers were used in the numerical study: Re 5 5.0 3 10 5 , Re 5 1.5 3 10 6 and Re 5 1.0 3 10 7 . There were two versions of the nozzle produced and, consequently, also simulated. One had a two-dimensional shape, with upper walls made of solid aluminum, and side walls of glass in order to allow optical visualization; the other nozzle was manufactured out of stainless steel and had the usual annular shape. Accordingly, the simulation assumed in the first case a two-dimensional flow, and in the second case an axisymmetric geometry. The physical flow conditions are discussed below.
2. Numerical approach The governing equations to be solved in the present simulations are the two-dimensional or axisymmetric compressible Navier–Stokes equations. They are, in a simplified vector form in general, body-fitted coordinates j and h in the weak conservation law form: ≠Q ≠(F 2 Fv ) ≠(G 2 Gv ) S ] 1 ]]] 1 ]]] 5 ] ≠t ≠j ≠h J
(1)
The above equations are given in more detail by, for example, Steger [4], and will not be repeated here. Eq. (1) is integrated in time by solving its semidiscrete form by means of modified Runge–Kutta (R– K) time stepping. The presently used two stage version of the R–K procedure can be written as
Q 1 5 Q n 2 a1 Dt * RsQ nd Q 2 5 Q n 2 a2 Dt * RsQ 1d Q
n11
5Q
(2)
2
with the residual R given by ≠(F 2 Fv ) ≠(G 2 Gv ) ≠Q ] 5 2 ]]] 2 ]]] 5 R ≠t ≠j ≠h
(3)
The coefficients a1 and a2 were optimized by von Lavante et al. [5] for maximum multigrid performance (damping of high frequencies). That optimization was, however, done using a linear hyperbolic model equation. In realistic applications, these coefficients worked fine for the inviscid Euler equations as well as for some viscous cases. The simple two stage R–K procedure was as efficient as the more frequently used four stage scheme, but required only one optimized coefficient, a1 5 0.42. Due to the complexity of the predicted flow, a simple numerical scheme with central spacial differences and artificial dissipation added explicitly was not suited for the present simulations. Therefore, the well known and proven Roe’s Flux Difference Algorithm [7] was subsequently employed. The numerical scheme is based on Roe’s Flux Difference Splitting in finite volume form, as developed by El-Miligui et al. [6]. In previous numerical predictions, this scheme was highly effective in providing accurate viscous results for wide ranges of Mach numbers. The reconstruction of the cell-centered variables to the cell-interface locations was done using a monotone interpolation of the primitive variables vector Q p 5s r, u, v, pd T . The interpolation slope was limited by an appropriate limiter, according to the previously published MUSCL type procedure (see, for example, [6]). The viscous fluxes Fv and Gv were centrally differenced. In the present work, the boundary conditions were implemented with the help of dummy (ghost) cells. At the subsonic inflow boundary, a locally onedimensional weakly non-reflective boundary condition, based on the isentropic theory of the Riemann problem, was used. The tangential velocity, incoming Riemann invariant, total pressure and entropy in the first ghost cell were specified. The outgoing Riemann invariant was extrapolated from the interior domain. At the subsonic outflow boundary, the static
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
pressure was specified, subject to the fluctuations allowed by the non-reflective treatment. The remaining variables were extrapolated. At solid walls, the velocities were anti-reflected, resulting in zero velocity on the boundary. The static pressure and density were reflected, resulting in zero gradients of these variables at the wall (adiabatic wall). Finally, at the interzonal boundaries, the block grids were overlapping by two cells, providing smooth transitions of the dependant variables Q due to the present MUSCL extrapolation in the projection phase.
3. Experimental approach
3.1. Laser light sheet procedure The laser light sheet procedure is a well-known method to visualize flow patterns in flow fields in general and in gas measuring and controlling units in particular. For that purpose, a light sheet is generated that illuminates a flow field to be examined in a thin plane layer (thickness: approx. 0.5 mm). Within this layer, the flow behavior can be observed using the tracer streaks of scattering particles. Fig. 1 shows the test setup of the above described procedure to record flow images by means of the laser light sheet. The flow in the test nozzle, equipped with optical inputs, is optically accessible for a camera through an inspection window. The laser light sheet is generated by the optical inputs on the top and bottom of the test nozzle. A vacuum blower generates the flow through the
3
test setup. The extent of the negative pressure downstream of the nozzle is adjusted by means of a bypass valve which is installed between the test nozzle and the blower. To visualize the air flow, an aerosol generator was used to add DEHS (diethyl sebacate) particles in sizes between 0.2 and 1.0 mm. When these particles suspended in the flow pass the light sheet, they scatter the laser light and are therefore visible. Within the light sheet, an image of the flow pattern appears that is recorded by a video camera through the inspection window. The analog output signal of the camera is digitized by a frame grabber board installed in a PC and then stored. Eventually, a digital post-processing of the images will be carried out. In this case, a nozzle-shaped rectangular channel is used. The light sheet is arranged in such a manner that the camera is able to record the flow along the axis of the nozzle.
3.2. Pressure measurements of nozzle flow In order to measure the static pressure of the flow along the nozzle wall, 18 pressure measuring locations, shown in Fig. 5, were distributed between the throat and the exit end of a nozzle with a diameter of the throat d th 510 mm. The individual pressure measuring points have been arranged parallel to the direction of the flow and alternate in two series set off by 908 (see Fig. 2). Each pressure measuring tap was threaded to allow the installation of a pressure sensor. The pressure was measured by a piezoresistive miniature precision pressure transducer. The sensor had an outside diameter of d52 mm and was integrated in a hollow bolt. It could therefore be inserted into any of the individual measuring points in a reproducible way. The openings not in use were hermetically plugged by proper Allen screws.
4. Results
Fig. 1. Schematic diagram of the test setup.
Initially, the exact flow character at the conditions given by the ISO 9300 Norm was not known. It was, therefore, decided to simulate the corresponding case numerically, hoping to obtain a better idea about the flow field. As the simulation results revealed surprisingly complex features of the flow, the simulation
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Fig. 2. Design drawing of the nozzle with the location of taps for pressure sensors.
was followed up by the experimental effort in order to validate the numerical data. First, the flow in the baseline nozzle shape, given in Fig. 3 including the computational grid, was computed. In this case, the fluid (air) was entering the nozzle at atmospheric stagnation conditions through the left boundary, with the total temperature T 0 5 293.14 K and total pressure P0 5 1013.25 mbar. The pressure difference across the nozzle was Dp 5 300 mbar. The outflow boundary is to the right, whereas the upper boundary is solid (wall) and the lower boundary represents the r50 symmetry line. The critical Reynolds number was in this case Re * 5 1.65 3 10 5 , which is at the lower limit given by the ISO 9300 Standard and corresponds to a Re ISO 5 1.31 3 10 5 . According to the real physics of the flow, the nozzle started from the zero flow condition (quiescent), the velocity being induced by the low pressure at the end of the nozzle. According to the simple quasi-one-dimensional theory, it was from the beginning obvious that the above pressure difference was too small to result in a smooth supersonic flow in the
entire nozzle. A system of complex shock waves causing a periodic flow separation at the wall developed. The separated bubbles detached from the wall and formed vortices that were convected downstream. The separated boundary layer had a noticeable blockage effect. The resulting pressure and density contours are shown in Fig. 4. The unsteady character of the flow is, of course, detrimental to the accuracy of the nozzle as a flow metering device. At this point, the authors decided to pursue two parallel investigations. First of all, an experiment had to be conceived to validate the above numerical results. Secondly, the numerical work was continued, trying to find a simple fix to the unsteadiness problem. Fig. 5 shows a diagram of the static pressure ps at four different locations in the nozzle as a function of the back pressure (pressure after the exit from the nozzle) pne . The exit pressures pne have been indicated as differential pressures with respect to the atmospheric pressure. The four locations of the static pressure measurement shown in Fig. 5 were chosen to be in front of the nozzle throat (positions 3 and 4), in the nozzle
Fig. 3. Computational grid for the baseline nozzle.
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
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Fig. 4. (a) Simulation results showing velocity contours; (b) simulation results showing static pressure; (c) simulation results showing density contours.
Fig. 5. Pressure measured at four locations in the nozzle. The locations are indicated in the small insert.
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throat itself (position 5) as well as in the outlet of the nozzle (position 6). As could be expected, the pressure in the nozzle throat asymptotically approaches an absolute static pressure of approx. ps 5 542 mbar at an atmospheric equilibrium pressure of patm 5995.5 mbar. This corresponds to a pressure ratio of ps /patm 50.544. This ratio is somewhat larger than the theoretical critical pressure ratio of ps /patm 50.528 given in the literature [8]. The reason for this discrepancy can be explained by noticing that the theory assumes one-dimensional, inviscid flow. As the results of the simulation clearly show, for example, in Fig. 4, the flow is of highly complex quasi-three-dimensional nature, with domains showing massive flow separation. The critical condition in the nozzle throat cannot be directly recognized as a straight isotach. Instead, the isotach is curved and located directly after the nozzle throat. At back pressures of pne 5125.5 mbar or higher, the static pressure ps at positions 3, 4 and 5 remains constant (see Fig. 5). However, the values of the absolute static pressure in front of the throat (positions 3 and 4) are at a higher level than the pressure in the
nozzle throat. According to the simple isentropic theory, they are supposed to behave this way due to their large cross sectional areas. The graph of the static pressure at a location downstream of the nozzle throat (position 6) is different, however. In the case of fairly high back pressures, the flow through the nozzle is subcritical, and the static pressure at position 6 will be higher than that in the nozzle throat because of the larger cross section. As the back pressure is further decreased, the critical conditions will be eventually reached in the throat, and therefore the flow will be supersonic at position 6, showing a static pressure lower than the critical pressure. Caution should be exercised, however; according to the numerical results, and confirmed by the experiments, purely supersonic conditions can not be maintained throughout the nozzle due to the relatively large back pressure. At a short distance downstream of the throat, a system of oblique shocks will form, causing an increase in the pressure. The supersonic flow will prevail only over short distances, for short periods of time (see below). Fig. 6 shows a comparison of the computed and
Fig. 6. Pressure along the wall of the nozzle wall.
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
measured absolute static pressures along the wall of the nozzle for Reynolds number Re * 5 1.5 3 10 5 . The outflow pressure ratio was in this case varied between pout /p0 5 0.95 and pout /p0 5 0.643. As pointed out above, at this Reynolds number, these outflow pressure ratios are too high to give smooth flow in the nozzle throat, resulting in unsteady flow with periodic unchoking of the nozzle. Contrary to the simple quasi-one-dimensional theory, the nozzle is not choked at the highest pressure ratio, indicating significant viscous effects in the diffusor. At all the higher pressure ratios, the nozzle is obviously choked. Considering the significant pressure fluctuations especially in the throat region, and the fact that in the experimental work, the time-wise averaging is different from the one used in the numerical analysis, the agreement is rather good. Principally, all graphs show similar characteristics. The flow is accelerated in the inlet of the nozzle, resulting in a reduction of the static pressure. The divergent part of the nozzle actually serves as a diffusor to accomplish the pressure recovery necessary for the relatively high exit pressure. However, it must be distinguished between three different cases. Case I is characterized by fully subsonic flow ( pne 5 50 mbar); the flow rate per area reaches its maximum, and the pressure its minimum, at the narrowest part of the nozzle, i.e. the nozzle throat (x516 mm). The diffuser reduces the Mach number, thus increasing the static pressure ps in the flow. Case II is reached for an exit pressure of approx. pne 5 130 mbar. This corresponds to a static pressure in the throat of ps 5 530 mbar. The flow will just become critical, and then follow the subcritical path to the exit. The pressure will subsequently recover the exit pressure of pne 5 130 mbar. Case III is finally obtained by further reducing the exit back pressure to pne 5 300 mbar. The velocity in the nozzle throat again reaches the speed of sound, with the corresponding critical pressure. Additionally, there is also supersonic flow in the beginning of the diverging part of the nozzle, indicated by the continuing
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decrease of the static pressure (x521 mm). This is accompanied by a corresponding increase of the flow Mach number, reaching values greater than the speed of sound. As shown in Figs. 4 and 11, the level of the exit pressure in Case III leads to the formation and shedding of vortices close to the wall. This is caused by a series of compression waves that lead to strong fluctuation of the static pressure at a given location on the wall. This effect is displayed by the rise of the static pressure ps after reaching its minimum at x521 mm. Any further decrease of the exit pressure leads to a shift of the minimum of the static pressure downstream. (x525.5 mm). An obvious method to achieve a smoother flow in the nozzle was to lower the backpressure. This, however, was deemed undesirable due to the significant increase in effort in generating it. Lastly, in order to validate the computations in the basic ‘short’ nozzle, the nozzle geometry was extended to include also a part of the duct into which the gas from the nozzle was flowing. In particular, it was necessary to prove that the flow instabilities were not caused by the outflow boundary condition. The geometry of this configuration can be seen in Fig. 7. Noteworthy is the long extension of the duct, eliminating any possibility of the outflow boundary influencing the flow in the nozzle itself. Fortunately for the investigators, the simulations in this geometry confirmed the previous findings, giving very similar results. The baseline geometry was also computed as a two-dimensional configuration (as opposed to the axisymmetric approach used in the work presented above) in order to allow a qualitative comparison with the light sheet flow visualization described above. Fig. 8 displays a detailed view of the boundary layer at the upper wall obtained in the present simulation at an axial location of approximately 38 mm after the throat. Compared to the picture obtained by the laser light sheet method (Fig. 9) at the same axial location, the fair level of qualitative agreement is apparent. The generation (shedding) of
Fig. 7. Computational grid for the extended nozzle.
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Fig. 8. Detailed view of the upper boundary layer.
Fig. 9. Laser light sheet picture of the 2-D nozzle.
vortices in the boundary layer due to the shockboundary layer interaction can be also seen in Fig. 10. Here, the timewise development of the flow, as simulated in the present work, is presented in terms of the velocity magnitude. The nozzles in the present work are to be used for gas metering, so that the mass flow through the throat is of particular interest. Theoretically, in a choked nozzle, it should be constant with time. However, due to the pressure waves travelling upstream, the throat in the smallest nozzle (Re * 5 1.0 3 10 5 ) is periodically unchocking, resulting in an intermittent decrease of the mass flow. The computed timewise history of the mass flow, non-dimensionalized by the theoretical (inviscid, quasione-dimensional, ideal nozzle) mass flow, is shown in Fig. 11a–f. It should be noted that this expression corresponds to an instantaneous value of the nozzle
discharge coefficient CD . The mass flow histories are displayed for the original ‘short’ configuration in Fig. 3 (Fig. 11a, c and e) and the extended ‘long’ configuration in Fig. 7 (Fig. 11b, d and f) for three different Reynolds numbers. Obviously, the two nozzle configurations give very similar results. Interestingly, the higher Reynolds numbers with the corresponding decrease in the boundary layer thickness result in more stable flow conditions. At the Reynolds number Re * 5 5.0 3 10 5 , the frequency and amplitude of the unsteady pressure waves is smaller, as indicated by the smaller mass flow fluctuations in Fig. 11c and d. At Reynolds numbers Re * 5 1.5 3 10 6 and higher, the flow in the throat remains choked at all times, as evidenced by the constant mass flow in Fig. 11e and f.
5. Conclusion In the present experimental and numerical study of the flow in critical nozzles used for gas metering, good agreement between the numerical and experimental results was achieved. Both approaches indicate that the flow in these devices is highly complex, with massively separated regions and strongly unsteady pressure waves. The pressure waves will propagate upstream and cause a periodic fluctuation of the mass flow in the throat and at the exit of the nozzles. For short periods of time the flow becomes subcritical with the corresponding decrease of mass flow. The unsteady behavior, shown in Fig. 11 as a plot of the mass flow at the nozzle throat as a
Fig. 10. (a) Computational results for the 2-D nozzle, displayed as velocity magnitude contours; (b) computational results for the 2-D nozzle, displayed as velocity magnitude contours; (c) computational results for the 2-D nozzle, displayed as velocity magnitude contours; (d) computational results for the 2-D nozzle, displayed as velocity magnitude contours.
E. von Lavante et al. / Measurement 29 (2001) 1 – 10
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Fig. 11. (a) Dimensionless mass flow as a function of time, ‘short’ nozzle; (b) dimensionless mass flow as a function of time, ‘long’ nozzle; (c) dimensionless mass flow as a function of time, ‘short’ nozzle; (d) dimensionless mass flow as a function of time, ‘long’ nozzle; (e) dimensionless mass flow as a function of time, ‘short’ nozzle; (f) dimensionless mass flow as a function of time, ‘long’ nozzle.
function of time, measured in seconds, is the result of the poorly matched back pressure to the shape of the nozzle. However, a purely supersonic flow in the nozzle would require an unrealistically low exit pressures. A compromise is needed in order to solve this problem. A major shortcoming of the present
numerical simulations is its two-dimensional character. Although no positive proof is available, the authors are certain that the real separation regions and other disturbances are not uniformly distributed around the circumference of the nozzles. The computed discharge coefficients are therefore underesti-
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mated and should be only used to indicate certain tendencies. A genuinely three-dimensional simulation is costly but unavoidable. Clearly, more work in this research area is required if recommendations regarding the improvement of accuracy of the critical nozzles are to be made.
[7]
[8] [9]
References ¨ [1] E. von Lavante, J. Gronner, Semi-implicit Schemes for Solving the Navier–Stokes Equations, 9th GAMM Conference, Lausanne, Notes on Numerical Fluid Mechanics, Vieweg, 1992. [2] E. von Lavante, A. El-Milligui, H. Warda, Simulation of Unsteady Flows Using Personal Computers, The 3rd International Congress on Fluid Mechanics, Cairo, 1990. [3] R.J.G. Norton, Prediction of the Installed Performance of 2-dimensional Exhaust Nozzles, AIAA Paper 87-0245. [4] J.L. Steger, Implicit, Finite-Difference Simulation of Flow about Arbitrary Geometries, AIAA Journal 16 (7) (1978) 679–686. [5] E. von Lavante, A. El-Miligui, F.E. Cannizzaro, H.A. Warda, Simple Explicit Upwind Schemes for Solving Compressible Flows, Proceedings of the 8th GAMM Conference on Numerical Methods in Fluid Mechanics, Delft, 1990. [6] A. El-Miligui, F. Cannizzaro, N.D. Melson, E. von Lavante, A Three-Dimensional Multigrid Multiblock Multistage Time Stepping Scheme for the Navier-Stokes Equations, Proceed-
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ings of the 9th GAMM Conference on Numerical Methods in Fluid Mechanics, Lausanne, 1991. P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Comp. Physics 43 (1981) 357–372. A.H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, John Wiley & Sons, 1953. ´ B. Nath, H. Dietrich, Compteur etalon automatise´ a` tubes de ´ Venturi fonctionnant en zone critique pour l’etalonnage de ´ ` ´ ` debitmetres ou de compteurs volumetrique, 8e Congres ´ ´ International de Metrologie Metrologie 97, 20.10.–23.10.97, Besanc¸on, France. International Standard ISO 9300, Measurement of Gas Flow by Means of Critical Flow Venturi Nozzles, 1st Edition, Beuth, Berlin, 1990. J.D. Anderson, Modern Compressible Flow, Mc Graw-Hill, New York, 1990. M. Ishibashi, M. Takamoto, Discharge Coefficient of Superaccurate Critical Nozzle at Pressurized Condition, Proccedings of the 4th International Symposium on Fluid Flow Measurement, Denver, USA, June 1999. H. Dietrich, B. Nath, E. von Lavante, High Accuracy Test Rig for Gas Flows from 0.01 m 3 / h up to 25 000 m 3 / h, Proceedings of the IMEKO-XV Congress, Osaka, Japan, June 1999. M. Ishibashi, M. Takamoto, Very Accurate Analytical Calculation of the Discharge Coefficient of Critical Venturi Nozzles with Laminar Boundary Layer, Proceedings of the FLUCOME97 Conference in Hayama, Japan, Sept. 1997. G. Wendt, Private Communication.