- Email: [email protected]
Simulations of transonic shock-tube flow with a model micro-cylinder in the driver
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 5 ( 2 0 0 7 ) 124–128
journal homepage: www.intl.elsevierhealth.com/journals/cmpb
Simulations of transonic shock-tube flow with a model micro-cylinder in the driver Yi Liu a,b,∗ , Mark A.F. Kendall b,c a b c
School of Mechanical Engineering, Southern Yangtze University, 1800 Lihu Road, Wuxi, Jiangsu Province 214122, PR China Department of Engineering Science, University of Oxford, Oxford OX2 6PE, United Kingdom Australian Institute for Bioengineering and Nanotechnology, The University of Queensland, Brisbane Qld 4072, Australia
a r t i c l e
i n f o
a b s t r a c t
Article history:
A unique hand-held needle-free powder injection system, using a transient shock-tube flow
Received 28 July 2006
to deliver powder genes and drugs into human skin for a wide range of treatments, has been
Received in revised form
proposed. In the development of such devices, a strong non-linear phenomenon, possibly
23 October 2006
shock process instead of unsteady expansion waves, was observed in the driver portion
Accepted 23 October 2006
of the shock-tube flow in the presence of a gas micro-cylinder. In this paper, we further investigate effects of a model micro-cylinder in the driver on the gas dynamics of a proto-
Keywords:
type clinical device numerically. To accurately simulate such complex shock-tube flows, an
Laminar
efficient numerical solver, MIFVS, is extended to incorporate with a transition-modified tur-
Separation
bulence model. Comparison with experimental measurements shows that the extended
Shock-tube
MIFVS accurately predicts pressure traces in both laminar and turbulent regimes. The
Simulation
separation zone due to a strong non-linear process is properly captured via such transition-
Transition
modified turbulence model. Numerical investigations and discoveries are presented and
Turbulence
discussed. © 2006 Elsevier Ireland Ltd. All rights reserved.
1.
Introduction
This study arises from the development of a hand-held needlefree powder pharmaceuticals delivery device (called biolistics), using a shock-tube-based system to accelerate genes and drugs [1,2]. In principle, this technology uses a transient high speed jet flow to accelerate micro-particles to a momentum high enough to penetrate the outermost layer of human skin or mucosal tissue and target the cells of interest. Fig. 1 shows a clinical biolistic device. The key components and principles are described elsewhere [3,4]. Numerical methods were implemented to simulate the complete operation of prototype clinical devices, in a simplest form and an ideal operation condition, with an emphasis on the overall system performance [1,2,4]. It has been demonstrated experimentally and numer-
ically that these prototype biolistic devices can deliver the particles with a desired velocity range and spatial distribution. In contrast to the simplest form and ideal operation condition, an important feature of the clinical device is the micro-cylinder, in which high-pressure helium gas is stored, together with a spacer, positioned co-axially inside the driver (marked in Fig. 1), referred as a non-ideal driver. The gasdynamic effects of a simplified configuration of this non-ideal driver have been previously investigated, which employed a hemisphere-cylinder obstacle instead [5]. It was found from pressure transducer measurements that the transient shocktube flow in driver strongly deviates from theory, particularly near the obstacle. In this paper, we are seeking to employ computational fluid dynamics approach to further explore the detailed flow phenomena numerically, especially boundary
∗ Corresponding author at: Department of Engineering Science, University of Oxford, Oxford OX2 6PE, United Kingdom. Tel.: +44 1865 274740. E-mail address: [email protected] (Y. Liu). 0169-2607/$ – see front matter © 2006 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2006.10.007
125
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 5 ( 2 0 0 7 ) 124–128
In present transition study, the damping function, f (as used in Eq. (1)), is modified to include the ratio 25/A+ , representing an intermittency function in the transition simulation:
f = 1 − exp −0.0115y+
25 A+
where the parameter A+ is set to a critical value (e.g. 300, in this study) in laminar flows to guarantee the turbulence viscosity, t , very small. The normal value of A+ t for fully turbulent boundary layer is 25. In the transition region, A+ is derived by:
Fig. 1 – Schematic of a hand-held needle-free biolistic device, configured for clinical trials.
layer transition and shock formation during the transient process. Numerically accurate prediction of gas dynamics in such biolistic devices, which involve different flow regime, strong non-linear shock process and shock–boundary layer interactions, is very demanding. Over the past decade, an accurate and efficient numerical solver, modified implicit flux vector splitting code (MIFVS), has been developed and applied to various flow analysis [2,6–10]. Numerical simulation with shock-tube-based biolistic devices also shows the MIFVS has obvious advantages in terms of accuracy, convergence and computing cost [2,10]. In experimental setup, no measurement other than pressure transducer traces were made due to the difficulties associated with a relatively small instrumental space (<10 mm in diameter) and a short operating duration (<500 s of interest) of the biolistic system. There is also no indication available from experimental data to identify the nature of the flow, whether laminar or turbulent. In the simplest form simulation aforementioned, either a laminar or a turbulent flow was assumed, with no transition taken into consideration. In this paper, we shall dynamically determine the onset of transition and incorporate with a low Reynolds number k–ε turbulence model. The objective of this study is to extend the MIFVS solver to automatically account for the transition process in transient transonic shock-tube flow, particularly in the region near a model micro-cylinder obstacle in the driver.
2.
Mathematical modeling
The formation of shock wave from reflected expansion waves within the simplified hemisphere-cylinder model in the driver is strongly non-linear, and also highly transient process. Together with the transition, which can be trigged by a local flow separation or a shock wave, numerically simulation of such non-linear problem is very challenging. In order to dynamically determine the onset of transition, a modified low Reynolds number k–ε models is proposed. Note that a general form for low Reynolds number k–ε models, proposed in various literatures so far, differs from the standard k–ε turbulence model in the use of damping function, f , which is only active close to solid walls and make it possible to solve k and ε down to viscous sub-layer [9]: f = 1 − exp(−0.0115y+ )
(2)
(1)
+ A+ = A+ t + (300 − At ) 1 − sin
Re − Re 2 tr 2
Retr
(3)
It is assumed that transition starts when the momentum thickness Reynolds number Re exceeds a critical Reynolds number Retr , and is completed when Re = 2Retr . Here, the critical Reynolds number is specified by the following equation [11]:
Retr = 163 + exp
F() −
F()I 6.91
(4)
in which I is the free-stream turbulence intensity, and
F() =
6.91 + 12.75 + 63.642
≤0
6.91 + 2.48 − 12.272
>0
(5)
is the Pohlhausen acceleration parameter, defined as: =
− 2 (∂p/∂x) Ube
(6)
where is the momentum thickness and is the molecular viscosity. The pressure gradient and the velocity Ube parallel to the solid wall (x-direction) are computed at the edge of the boundary layer. This form of transition model can be readily implemented into the low Reynolds number k–ε model. The damping function, Eqs. (1) or (2), can be dynamically evaluated in the turbulent or transient regimes. In addition to the continuous phase equations for the laminar-turbulent flow, a discrete particle phase (diluted particles in a micro-sized form) is solved in a Lagrangian frame of reference. The dispersion of particles due to turbulence in the gas phase can be predicted using the particle cloud model, such as models proposed by Litchford and Jeng [12], Baxter and Smith [13]. However, computational cost for the particle cloud model in the current transient transition simulations is prohibitive. Moreover, for the drug delivery applications only the mean impact velocity of particles is of great importance in the determination of the penetration depth [14]. Therefore, a simplified approach, based on the mean velocity instead of instantaneous turbulent velocity for particle tracking and coupled discrete-phase calculations, is adopted [8].
126
3.
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 5 ( 2 0 0 7 ) 124–128
Numerical solution
An implicit solver for the Reynolds averaged Navier–Stokes equations, MIFVS, has been previously developed, which uses the spectral radii technique to simplify the calculation, and at the same time avoids the approximate-factorization (AF) to increase the time step and stability [6]. The matrix operation is not needed, therefore, the amount of computing work in each time step can be significantly reduced. The MIFVS simulations have proven to be adequately efficient in various turbomachinery applications [6–9] and shock-tube problems [2,10], while maintaining a high level of robustness and accuracy. We now apply this code, incorporating the transitionmodified turbulence model, to simulate transient shock-tube flows with a model micro-cylinder obstacle in a shock-tube driver in detail. Together with Eqs. (3)–(6), the turbulence model is solved in a similar manner as in the MIFVS code.
4.
Fig. 3 – The pressure histories.
Results and discussion
The configuration under numerical concern, illustrated in Fig. 2, is the same geometry as experimental studies [5]. The method for mesh generation and the specification for boundary conditions are similar to other shock-tube-based system simulations [2,10]. The presented results have been verified to be grid-independent with a maximum discrepancy less than 1% for the most important parameters. A grid of total 43,950 cells is discretised over the computational domain with y+ about 2–10 near the wall. To recognize characteristics of the proposed transition model in a transient transonic shock-tube flow, initially laminar, turbulent and transition-modified turbulent models are simulated, respectively. The calculated pressure histories, compared with measurements, are shown in Fig. 3, with the time zero denoting as instant rupture of diaphragm. The experimental setup and pressure traces were reported in Ref. [5]. It is observed that the turbulent solution fails to capture pressure drop in Position T1 (labeled as in the figure) and Position T2 (ˇ). Physically, the pressure drop represents a complex flow phenomenon in the nearby position, involving a non-linear shock formation, its interaction with the boundary layer and subsequent separation. It will be illustrated
Fig. 4 – The Mach number histories.
in two-dimensional figures, too. Like laminar solutions, the transition-modified turbulence model adequately captures the pressure drop, but with a more accuracy. Comparison of pressure histories between the experimental traces and simulations for Positions T1 and T2 in the driver indicates that the non-linear shock process and transition play an important role. Simulations also discover that compared with an ideal shock-tube, unlike in the driver, there are no obvious differences in the driven section (e.g. the pressure histories at T4 , not shown here) for the duration of interest. The simulated Mach numbers at the center of the shocktube exit-plane (labeled as TPitot in Fig. 2), together with an analytical value of 0.96, and experimental derived Mach number (1.03 ± 0.12, the mean velocity ± standard deviation), are displayed in Fig. 4. The discrepancy between predicted and measured Pitot pressure and the resulting Mach number is consistent with results from other shock-tube-based devices. This is largely due to the presence of the Pitot probe
Fig. 2 – Configuration of the simplified device and experimental setup.
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 5 ( 2 0 0 7 ) 124–128
127
Initial comparison with experimental pressure traces shows that the extended MIFVS with the proposed transitionmodified turbulence model simulates the transient transonic shock-tube flow with a better agreement than corresponding calculation with turbulence model, or laminar solutions. The separation due to a strong non-linear process is accurately captured. However, this separation bubble is not developed to the driven section during a period of 500 s, which is typical time designed for powdered pharmaceuticals (gene/drug) delivery. It is demonstrated that the simplified non-ideal driver owing to the presence of the model micro-cylinder has no significant effects on the delivery of powdered pharmaceuticals in the clinical biolistic devices. These simulations provide us a useful guideline for the design and optimization of the practical hand-held biolistic devices.
Acknowledgements
Fig. 5 – Instantaneous Mach number contours at a time of 180 s (a), and 250 s (b).
(sized 2 mm in diameter, compared with 10 mm at the shocktube exit). However, the transition-modified turbulence model gives a best prediction. Fig. 5 shows instantaneous Mach number contours at two typical times after diaphragm rupture, representing the arriving and passing of the reflected expansion wave at Positions T1 and T2 (shown in Fig. 2), respectively. The reflected expanded waves propagate through a curved pathway. As a result of the effective area changes, the shock wave is formed from the coalescence of unsteady expansion waves just around the corner as label in the figure. These non-linear expansion and shock waves are further interacted with the wall boundary layer. Subsequently the separation occurs, first detected from T1 (at a time of 180 s, Fig. 5a) then extended to T2 (at a time of 250 s, Fig. 5b), which is consistent with the measured and simulated pressure evolutions at Positions of T1 and T2 (shown in Fig. 3). It is also shown that the separation zone within the driver section has a trivial effect on the gas flow downstream of the driven section. With simulated non-linear process in the simplified driver, the system can be then properly designed and optimized, ensuring that the particles are delivered before the arrival of the distorted reflected expansion and shock waves, thereby, within an un-effected quasi-steady quasi-one dimensional delivery window.
5.
Conclusion
A model micro-cylinder obstacle in the shock-tube driver (simplified driver), an important feature of hand-held biolistic devices, is investigated numerically. An efficient MIFVS solver, together with a proposed transition-modified k–ε turbulence model, is extended to interrogate the non-linear shock formation, transition and shock/boundary layer interaction.
The authors would like to thank Prof. B.J. Bellhouse, Dr. F. Cornhill and Dr. G. Costigan for their technical insights and encouragements. Dr. F. Carter, Dr. N.K. Truong and Mr. M.C. Marrion are gratefully acknowledged for their experimental work of prototype biolistic devices.
references
[1] Y. Liu, M.A.F. Kendall, N.K. Truong, B.J. Bellhouse, Numerical and experimental analysis of a high speed needle-free powdered vaccines delivery device, in: Proceedings of the 20th AIAA Applied Aerodynamics Conference, AIAA-2002-2807, St. Louis, MO, USA, 2002. [2] Y. Liu, M.A.F. Kendall, Numerical simulation of heat transfer from a transonic jet impinging on skin for needle-free powdered drug and vaccine delivery, J. Mech. Eng. Sci.: Proc. Inst. Mech. Eng. Pt. C 218 (11) (2004) 1373–1383. [3] M.A.F. Kendall, The delivery of particulate vaccines and drugs to human skin with a practical, hand-held shock-tube-based system, Shock Waves J. 12 (1) (2002) 22–30. [4] Y. Liu, M.A.F. Kendall, Numerical study of a transient gas and particle flow in a high-speed needle-free ballistic particulate vaccine delivery system, J. Mech. Med. Biol. 4 (4) (2004) 559–578. [5] M.C. Marrion, M.A.F. Kendall, Y. Liu, The gas-dynamic effects of a hemisphere-cylinder obstacle in a shock-tube driver, Exp. Fluids 28 (3) (2005) 319–327. [6] Y. Liu, Numerical method of three-dimensional viscous flow in turbomachinery multistage environment, PhD Thesis, Xi’an Jiaotong University, China, 1996. [7] Y. Liu, Q. Cao, Y.M. Xiang, Numerical simulation of viscous flows in transonic diffuser, J. Hydrodynam. Ser. A 12 (1) (1997) 33–37. [8] Y. Liu, M.A.F. Kendall, B.J. Bellhouse, An efficient implicit finite difference scheme for transonic flow, in: Proceedings of the 32nd AIAA Fluid Dynamics Conference and Exhibit, AIAA-2002-2955, St. Louis, MO, USA, 2002. [9] Y. Liu, Simulations of shock/boundary layer interactions over high lift turbine blades, in: Proceedings of the ASME Turbo Expo 2003, ASME-GT-2003-38723, Atlanta, GA, USA, 2003. [10] Y. Liu, B.J. Bellhouse, Prediction of jet flows in the supersonic nozzle and diffuser, Int. J. Numer. Method Fluids 47 (2005) 1147–1155.
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c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 5 ( 2 0 0 7 ) 124–128
[11] B.J. Abu-Ghannam, R. Shaw, Natural transition of boundary layers—the effects of turbulence, pressure gradient, and flow history, J. Mech. Eng. Sci. 22 (5) (1980) 213–228. [12] R.J. Litchford, S.M. Jeng, Efficient statistical transport model for turbulent particle dispersion in sprays, AIAA J. 29 (1991) 1443.
[13] L.L. Baxter, P.J. Smith, Turbulent dispersion of particles: the STP model, Energy Fuels 7 (1993) 852–859. [14] J. Suh, M. Dawson, J. Hanes, Real-time multiple-particle tracking: applications to drug and gene delivery, Adv. Drug Deliv. Rev. 57 (2005) 63–78.
journal homepage: www.intl.elsevierhealth.com/journals/cmpb
Simulations of transonic shock-tube flow with a model micro-cylinder in the driver Yi Liu a,b,∗ , Mark A.F. Kendall b,c a b c
School of Mechanical Engineering, Southern Yangtze University, 1800 Lihu Road, Wuxi, Jiangsu Province 214122, PR China Department of Engineering Science, University of Oxford, Oxford OX2 6PE, United Kingdom Australian Institute for Bioengineering and Nanotechnology, The University of Queensland, Brisbane Qld 4072, Australia
a r t i c l e
i n f o
a b s t r a c t
Article history:
A unique hand-held needle-free powder injection system, using a transient shock-tube flow
Received 28 July 2006
to deliver powder genes and drugs into human skin for a wide range of treatments, has been
Received in revised form
proposed. In the development of such devices, a strong non-linear phenomenon, possibly
23 October 2006
shock process instead of unsteady expansion waves, was observed in the driver portion
Accepted 23 October 2006
of the shock-tube flow in the presence of a gas micro-cylinder. In this paper, we further investigate effects of a model micro-cylinder in the driver on the gas dynamics of a proto-
Keywords:
type clinical device numerically. To accurately simulate such complex shock-tube flows, an
Laminar
efficient numerical solver, MIFVS, is extended to incorporate with a transition-modified tur-
Separation
bulence model. Comparison with experimental measurements shows that the extended
Shock-tube
MIFVS accurately predicts pressure traces in both laminar and turbulent regimes. The
Simulation
separation zone due to a strong non-linear process is properly captured via such transition-
Transition
modified turbulence model. Numerical investigations and discoveries are presented and
Turbulence
discussed. © 2006 Elsevier Ireland Ltd. All rights reserved.
1.
Introduction
This study arises from the development of a hand-held needlefree powder pharmaceuticals delivery device (called biolistics), using a shock-tube-based system to accelerate genes and drugs [1,2]. In principle, this technology uses a transient high speed jet flow to accelerate micro-particles to a momentum high enough to penetrate the outermost layer of human skin or mucosal tissue and target the cells of interest. Fig. 1 shows a clinical biolistic device. The key components and principles are described elsewhere [3,4]. Numerical methods were implemented to simulate the complete operation of prototype clinical devices, in a simplest form and an ideal operation condition, with an emphasis on the overall system performance [1,2,4]. It has been demonstrated experimentally and numer-
ically that these prototype biolistic devices can deliver the particles with a desired velocity range and spatial distribution. In contrast to the simplest form and ideal operation condition, an important feature of the clinical device is the micro-cylinder, in which high-pressure helium gas is stored, together with a spacer, positioned co-axially inside the driver (marked in Fig. 1), referred as a non-ideal driver. The gasdynamic effects of a simplified configuration of this non-ideal driver have been previously investigated, which employed a hemisphere-cylinder obstacle instead [5]. It was found from pressure transducer measurements that the transient shocktube flow in driver strongly deviates from theory, particularly near the obstacle. In this paper, we are seeking to employ computational fluid dynamics approach to further explore the detailed flow phenomena numerically, especially boundary
∗ Corresponding author at: Department of Engineering Science, University of Oxford, Oxford OX2 6PE, United Kingdom. Tel.: +44 1865 274740. E-mail address: [email protected] (Y. Liu). 0169-2607/$ – see front matter © 2006 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2006.10.007
125
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 5 ( 2 0 0 7 ) 124–128
In present transition study, the damping function, f (as used in Eq. (1)), is modified to include the ratio 25/A+ , representing an intermittency function in the transition simulation:
f = 1 − exp −0.0115y+
25 A+
where the parameter A+ is set to a critical value (e.g. 300, in this study) in laminar flows to guarantee the turbulence viscosity, t , very small. The normal value of A+ t for fully turbulent boundary layer is 25. In the transition region, A+ is derived by:
Fig. 1 – Schematic of a hand-held needle-free biolistic device, configured for clinical trials.
layer transition and shock formation during the transient process. Numerically accurate prediction of gas dynamics in such biolistic devices, which involve different flow regime, strong non-linear shock process and shock–boundary layer interactions, is very demanding. Over the past decade, an accurate and efficient numerical solver, modified implicit flux vector splitting code (MIFVS), has been developed and applied to various flow analysis [2,6–10]. Numerical simulation with shock-tube-based biolistic devices also shows the MIFVS has obvious advantages in terms of accuracy, convergence and computing cost [2,10]. In experimental setup, no measurement other than pressure transducer traces were made due to the difficulties associated with a relatively small instrumental space (<10 mm in diameter) and a short operating duration (<500 s of interest) of the biolistic system. There is also no indication available from experimental data to identify the nature of the flow, whether laminar or turbulent. In the simplest form simulation aforementioned, either a laminar or a turbulent flow was assumed, with no transition taken into consideration. In this paper, we shall dynamically determine the onset of transition and incorporate with a low Reynolds number k–ε turbulence model. The objective of this study is to extend the MIFVS solver to automatically account for the transition process in transient transonic shock-tube flow, particularly in the region near a model micro-cylinder obstacle in the driver.
2.
Mathematical modeling
The formation of shock wave from reflected expansion waves within the simplified hemisphere-cylinder model in the driver is strongly non-linear, and also highly transient process. Together with the transition, which can be trigged by a local flow separation or a shock wave, numerically simulation of such non-linear problem is very challenging. In order to dynamically determine the onset of transition, a modified low Reynolds number k–ε models is proposed. Note that a general form for low Reynolds number k–ε models, proposed in various literatures so far, differs from the standard k–ε turbulence model in the use of damping function, f , which is only active close to solid walls and make it possible to solve k and ε down to viscous sub-layer [9]: f = 1 − exp(−0.0115y+ )
(2)
(1)
+ A+ = A+ t + (300 − At ) 1 − sin
Re − Re 2 tr 2
Retr
(3)
It is assumed that transition starts when the momentum thickness Reynolds number Re exceeds a critical Reynolds number Retr , and is completed when Re = 2Retr . Here, the critical Reynolds number is specified by the following equation [11]:
Retr = 163 + exp
F() −
F()I 6.91
(4)
in which I is the free-stream turbulence intensity, and
F() =
6.91 + 12.75 + 63.642
≤0
6.91 + 2.48 − 12.272
>0
(5)
is the Pohlhausen acceleration parameter, defined as: =
− 2 (∂p/∂x) Ube
(6)
where is the momentum thickness and is the molecular viscosity. The pressure gradient and the velocity Ube parallel to the solid wall (x-direction) are computed at the edge of the boundary layer. This form of transition model can be readily implemented into the low Reynolds number k–ε model. The damping function, Eqs. (1) or (2), can be dynamically evaluated in the turbulent or transient regimes. In addition to the continuous phase equations for the laminar-turbulent flow, a discrete particle phase (diluted particles in a micro-sized form) is solved in a Lagrangian frame of reference. The dispersion of particles due to turbulence in the gas phase can be predicted using the particle cloud model, such as models proposed by Litchford and Jeng [12], Baxter and Smith [13]. However, computational cost for the particle cloud model in the current transient transition simulations is prohibitive. Moreover, for the drug delivery applications only the mean impact velocity of particles is of great importance in the determination of the penetration depth [14]. Therefore, a simplified approach, based on the mean velocity instead of instantaneous turbulent velocity for particle tracking and coupled discrete-phase calculations, is adopted [8].
126
3.
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 5 ( 2 0 0 7 ) 124–128
Numerical solution
An implicit solver for the Reynolds averaged Navier–Stokes equations, MIFVS, has been previously developed, which uses the spectral radii technique to simplify the calculation, and at the same time avoids the approximate-factorization (AF) to increase the time step and stability [6]. The matrix operation is not needed, therefore, the amount of computing work in each time step can be significantly reduced. The MIFVS simulations have proven to be adequately efficient in various turbomachinery applications [6–9] and shock-tube problems [2,10], while maintaining a high level of robustness and accuracy. We now apply this code, incorporating the transitionmodified turbulence model, to simulate transient shock-tube flows with a model micro-cylinder obstacle in a shock-tube driver in detail. Together with Eqs. (3)–(6), the turbulence model is solved in a similar manner as in the MIFVS code.
4.
Fig. 3 – The pressure histories.
Results and discussion
The configuration under numerical concern, illustrated in Fig. 2, is the same geometry as experimental studies [5]. The method for mesh generation and the specification for boundary conditions are similar to other shock-tube-based system simulations [2,10]. The presented results have been verified to be grid-independent with a maximum discrepancy less than 1% for the most important parameters. A grid of total 43,950 cells is discretised over the computational domain with y+ about 2–10 near the wall. To recognize characteristics of the proposed transition model in a transient transonic shock-tube flow, initially laminar, turbulent and transition-modified turbulent models are simulated, respectively. The calculated pressure histories, compared with measurements, are shown in Fig. 3, with the time zero denoting as instant rupture of diaphragm. The experimental setup and pressure traces were reported in Ref. [5]. It is observed that the turbulent solution fails to capture pressure drop in Position T1 (labeled as in the figure) and Position T2 (ˇ). Physically, the pressure drop represents a complex flow phenomenon in the nearby position, involving a non-linear shock formation, its interaction with the boundary layer and subsequent separation. It will be illustrated
Fig. 4 – The Mach number histories.
in two-dimensional figures, too. Like laminar solutions, the transition-modified turbulence model adequately captures the pressure drop, but with a more accuracy. Comparison of pressure histories between the experimental traces and simulations for Positions T1 and T2 in the driver indicates that the non-linear shock process and transition play an important role. Simulations also discover that compared with an ideal shock-tube, unlike in the driver, there are no obvious differences in the driven section (e.g. the pressure histories at T4 , not shown here) for the duration of interest. The simulated Mach numbers at the center of the shocktube exit-plane (labeled as TPitot in Fig. 2), together with an analytical value of 0.96, and experimental derived Mach number (1.03 ± 0.12, the mean velocity ± standard deviation), are displayed in Fig. 4. The discrepancy between predicted and measured Pitot pressure and the resulting Mach number is consistent with results from other shock-tube-based devices. This is largely due to the presence of the Pitot probe
Fig. 2 – Configuration of the simplified device and experimental setup.
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 5 ( 2 0 0 7 ) 124–128
127
Initial comparison with experimental pressure traces shows that the extended MIFVS with the proposed transitionmodified turbulence model simulates the transient transonic shock-tube flow with a better agreement than corresponding calculation with turbulence model, or laminar solutions. The separation due to a strong non-linear process is accurately captured. However, this separation bubble is not developed to the driven section during a period of 500 s, which is typical time designed for powdered pharmaceuticals (gene/drug) delivery. It is demonstrated that the simplified non-ideal driver owing to the presence of the model micro-cylinder has no significant effects on the delivery of powdered pharmaceuticals in the clinical biolistic devices. These simulations provide us a useful guideline for the design and optimization of the practical hand-held biolistic devices.
Acknowledgements
Fig. 5 – Instantaneous Mach number contours at a time of 180 s (a), and 250 s (b).
(sized 2 mm in diameter, compared with 10 mm at the shocktube exit). However, the transition-modified turbulence model gives a best prediction. Fig. 5 shows instantaneous Mach number contours at two typical times after diaphragm rupture, representing the arriving and passing of the reflected expansion wave at Positions T1 and T2 (shown in Fig. 2), respectively. The reflected expanded waves propagate through a curved pathway. As a result of the effective area changes, the shock wave is formed from the coalescence of unsteady expansion waves just around the corner as label in the figure. These non-linear expansion and shock waves are further interacted with the wall boundary layer. Subsequently the separation occurs, first detected from T1 (at a time of 180 s, Fig. 5a) then extended to T2 (at a time of 250 s, Fig. 5b), which is consistent with the measured and simulated pressure evolutions at Positions of T1 and T2 (shown in Fig. 3). It is also shown that the separation zone within the driver section has a trivial effect on the gas flow downstream of the driven section. With simulated non-linear process in the simplified driver, the system can be then properly designed and optimized, ensuring that the particles are delivered before the arrival of the distorted reflected expansion and shock waves, thereby, within an un-effected quasi-steady quasi-one dimensional delivery window.
5.
Conclusion
A model micro-cylinder obstacle in the shock-tube driver (simplified driver), an important feature of hand-held biolistic devices, is investigated numerically. An efficient MIFVS solver, together with a proposed transition-modified k–ε turbulence model, is extended to interrogate the non-linear shock formation, transition and shock/boundary layer interaction.
The authors would like to thank Prof. B.J. Bellhouse, Dr. F. Cornhill and Dr. G. Costigan for their technical insights and encouragements. Dr. F. Carter, Dr. N.K. Truong and Mr. M.C. Marrion are gratefully acknowledged for their experimental work of prototype biolistic devices.
references
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