Shock Waves (2008) 18:205–212 DOI 10.1007/s00193-008-0155-9
ORIGINAL ARTICLE
Numerical simulation of blast flow fields induced by a high-speed projectile Xiaohai Jiang · Zhihua Chen · Baochun Fan · Hongzhi Li
Received: 19 February 2008 / Accepted: 13 June 2008 / Published online: 8 July 2008 © Springer-Verlag 2008
Abstract Numerical investigations on the launch process of a projectile in a nearly realistic situation have been performed in this article. The Arbitrary Lagrangian–Eulerian (ALE) of Euler equations is solved by the AUSMDV scheme and the dynamic chimera grid technique are used for describing the moving of the projectile. Based on our numerical results, the muzzle blast flow field of the transient launch process of a projectile at a relative high Mach number of 3.0 has been visualized numerically, and the prominent characteristics including the propagation of first and second blast waves, the generation of bow shock wave and moving of the projectile, etc. have been discussed in detail. Keywords Muzzle flow · Blast wave · AUSMDV scheme · Projectile launch PACS 47.40.-X · 47.11.Df · 47.15.K-
1 Introduction Investigations on blast flows have attracted many attentions for a long time due to its importance not only for understanding of dynamics of blast waves in scientific research but also for the design of devices used for the generation or attenuation of explosion in engineering industries. Some representative examples are gun-firing muzzle blast [1,2], suppression of sonic booms for hypersonic vehicles [3] and explosion Communicated by A. Merlen. X. Jiang · Z. Chen (B) · B. Fan · H. Li State Key Laboratory of Transient Physics, Nanjing University of Science and Technology, 210094 Nanjing, China e-mail:
[email protected]
venting technology used for protecting industrial plants from internal explosions [4]. One typical blast flow is the muzzle jet flow induced by the supersonic projectile. These kind of muzzle blasts are very complex and usually characterized mainly by two blast waves, two jet flows, and their interactions between the projectile. The first blast is caused by the precursor shock wave ahead of the projectile and the second is developed due to the expansion of high propellant gas behind the projectile. Many experimental shadowgraphs have visualized these characters [2,5]. On the basis of precise experimental visualizations, Merlen et al. [5] developed the similarity rules of muzzle wave propagation with a particular application to the case of quasi-steady and sonic muzzle conditions of air intakes interference problem for military aircrafts. Recent advances in computational fluid dynamics (CFD) techniques have made the numerical simulation of such complex flow fields become a highly effective alternative. Many pioneer investigations on muzzle flow with CFD were performed under some simplified conditions, and some phenomena of the blast flow field were described successfully. Using a second-order upwind scheme, the Euler equations were solved for simulating the muzzle blast flow fields by Jiang et al. [6]. In their work, the projectile is predigested as a cylinder and the flow is initiated as the precursor shock wave begins from the exit of the shock tube. The blast flow fields and wave dynamics processes are demonstrated and discussed in detail. Cayzac et al. predicted the muzzle flow with muzzle brakes [7] based on two-dimensional Euler equations and TVD scheme. And their boundary in the barrel was computed based on the characteristics method according to a pull or push piston analogy. For developing CFD modeling and validation of the CFD codes in simulating the nearfield wave propagation, Cler et al. [8] used two CFD codes, Fluent 6.1 (inviscid solver) and the Discontinuous Galerkin
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code in a comparison to simulate the muzzle flow without the moving projectile. Their numerical results were compared to the experimental shadowgraph (7.62 mm NATO rifle G3), and showed that the precursor flow matched the experimental well, but the main propellant flow did not match as well. They also found that the discontinuous solvers can model blast with courser gird adaptation than standard solvers. Dayan et al. [9] divided the muzzle flow calculation into two stages. First, the pressure and temperature, depended by the projectile motion were obtained via the internal ballistics calculation software package IBHVG2, and then, the projectile movement in the outer was simulated via commercial CFD-FASTRAN finite volume solver package. Very good agreements obtained for the precursor flow field with the predicted and the experimental results in first and second stage, respectively. Using Roe scheme, the authors of present study presented a numerical results of Euler equations for the study of the blast waves of a gun muzzle [10]. The complicated transient phenomena during the launch were disclosed and the generation, development and their coupling between the first air precursor shock wave and the second blasts were discussed in detail. However, the initial ratio of pressure behind the projectile to that in front of it during launching was limited to a critical value due to the carbuncle phenomenon. In this article, to capture the flow fields with strong blast fields and simulate the high Mach number of projectile, the AUSMDV scheme [11] is chosen for solving the Euler equations in ALE forms. The launch process is simulated numerically in a nearly realistic case. The projectile is initialized at the bottom of the tube and accelerated under the propulsion of very high pressure propellant gas behind the projectile. Based on our numerical results, the muzzle blast phenomena generated by a high speed projectile with Mach number of about 3.0 were visualized numerically and discussed in detail.
2 Numerical method 2.1 Governing equation Assuming the viscous effects and chemical reactions are negligible in current study. For a perfect gas, the Euler equations in a general deformation control volume can be expressed as (i.e. ALE equation [12]) ∂ Ud + (F−Uw) · nds =0 (1) ∂t (t)
S(t)
where U = [ρ, ρV , ρ E t ]T , F = U V + G, G = [0, p I, pV ]T . ρ, p is the density and pressure, respectively, V is the fluid velocity vector and E t is total energy per unit mass. S(t) is the surface which encloses the time dependent volume
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Fig. 1 Schematic of two grids system
(t), and n is the outward unit vector normal to the boundary S(t), I is the unit tensor and w denotes the surface velocity of the time dependent volume (t). In Eq. (1), w = V corresponds to the Lagrangian system, while w = 0 corresponds to the Eulerian one. In general, the Eq. (1) was called as the arbitrary Lagrangian–Eulerian form of the conservation system (ALE equations). 2.2 AUSMDV scheme In order to simulate the relative movement between projectile and gun tube, especially after its moving out of the tube, two conventional Cartesian grids system with uniform spatial resolution were employed and is shown in Fig. 1. The main background mesh is laboratory-fixed and the other moves with the projectile. The AUSM scheme [13] has the advantage of capturing shock waves efficiently, avoiding complicated matrix operations and differentiation of fluxes. In addition, it is very simple to construct and more efficient than other methods in dealing with a large set of equations in hyperbolic systems such as those described above. For AUSM scheme, Eq. (1) can be expressed as ∂ (2) Ud + F · nds =0 ∂t (t) S (t) where F is defined as F = (V − w) Q − P
(3)
where Q = [ρ, ρv, ρ H ]T , P = [0, − p I, − pw]T . H is the total enthalpy, and can be written as (4) H = Et + p ρ Equation (2) represents the conservation of mass, momentum and energy. For the high densities of gas generated during propellant combustion, the Noble–Abel equation can be used to account for the state of propellant gases. γ −1 ρe (5) p= 1 − bρ where e = E t − 21 V · V , is the specific internal energy. γ is the ratio of specific heats, and b refers to the co-volume of
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the gas. Equation (2) was discretized on the time dependent control volume, and can be written as n+1
Q
Vn = n+1 V
N 1 Fi Si t Q − n V
n
(6)
i=1
where the superscript n denotes the previous time and n1 is the solution time. N is the total number of the surfaces covering the control volume. Si refers to the ith surface area, and Fi is its normal flux. In the program the two-stage Rung–Kutta scheme was applied. Considering the control volume interface as one dimensional Riemann solution, the interface flux is calculated using AUSM scheme on a moving grid, for 1D case ⎛ F
1 i+ 2
⎟ ⎜ 1 2 1 2 2 ⎟ =⎜ ⎝ ( 2 + s f )(ρu )V + ( 2 − s f )(ρu ) D ⎠ (ρu) 1 Ht ⎛ 2 ⎞ 0 ⎜ pˆ 1 ⎟ ⎟ +⎜ ⎝ 2 ⎠ ( pw) 1
And the pressure splitting is pˆ
⎞
(ρu) 1
Fig. 2 Schematic of geometry and computational domain
1 i+ 2
pˆ i+ =
(7)
pˆ i−
2
2 − ρi+1 (ρu) 1 = uˆ i+ ρi + uˆ i+1
(ρu 2 )
D and V denote the term AUSMV, respectively [12],
ρu 2
in AUSMD and
− (ρu 2 )V = uˆ i+ (ρu)i + uˆ i+1 (ρu)i+1 (9) 1 (ρu 2 ) D = (ρu) 1 (u i + u i+1 )−(ρu) 1 (u i+1 − u i ) (10) 2 2 2
s f is a switch as a function of the local pressure gradient, | pi+1 − pi | 1 min(1.0, 10.0 ) 2 min( pi , pi+1 )
(11)
The velocity splitting within the AUSMDV is, ⎧ +ai )2 ⎨ αi (uˆ i4c − m ⎩ uˆ i +|uˆ i | 2
− uˆ i+1
m
⎪ ⎩p
− uˆ i+1 i+1 u i+1
uˆ i+1 am
if
(15)
|uˆ i+1 | cm
≤ 1, (16)
otherwise
2.3 Computational example
(ρu 2 )
=
m
+ ⎪ ⎩ p uˆ i otherwise i uˆ i ⎧ (uˆ +a )2 ⎪ ⎨ pi+1 i+14c2 i+1 · 2.0 +
(8)
2
uˆ i+
(14)
where a is the fluid sound speed and cm = max(ai , ai+1 ).uˆ denotes the fluid velocity relative to the interface velocity representing the effect of mesh deformation.
where (ρu) 1 is the AUSMDV mass flux,
sf =
=
− = p 1 = pi+ + pi+1 i+ 2 ⎧ |uˆ | +ai )2 ⎪ ⎨ pi (uˆ i4c if cmi ≤ 1, · 2.0 − auˆmi 2
uˆ i +|uˆ i | 2
+
uˆ i +|uˆ i | 2
if
|uˆ i | cm
(12)
otherwise
⎧ uˆ i+1 −|uˆ i+1 | (uˆ i+1 +ai+1 )2 ⎪ α − + − i+1 ⎪ 4cm 2 ⎪ ⎪ ⎨ uˆ i+1 −|uˆ i+1 | |uˆ | = if ci+1 ≤ 1, 2 m ⎪ ⎪ ⎪ ⎪ ⎩ uˆ i+1 −|uˆ i+1 | otherwise 2
Here αi =
≤ 1,
2( p/ρ )i , ( p/ρ )i +( p/ρ )i+1
αi+1 =
2( p/ρ )i+1 . ( p/ρ )i +( p/ρ )i+1
(13)
To simulate the launch process in a relative realistic condition, the projectile is put initially near the bottom of the gun tube as shown in Fig. 2. The original point of Cartesian coordinate is set at the muzzle and the x-coordinate is the same with the central axis of the tube. The motion of projectile obeys Newton’s second law: du p = A Pb − P f dt where m p is the mass of the projectile, A is the cross sectional area of the projectile, Pb and P f is the pressure of the projectile back and front side, respectively. When the projectile is inside the tube, the friction between the projectile and the tube is assumed to be negligible. The tube inner diameter, d = 7.62 mm, and its length l = 25d. The length of the projectile L = 2.5d, its diameter is the same with the inner diameter of the tube, and its mass is equal to 7.9 g. The initial inner pressure at the bottom of the projectile is set as pin = 3600 p0 , where p0 refers to the ambient air pressure under standard conditions. The gun propellant JA2 is chosen to be the fuel for our simulation. The approximate thermodynamic properties of its gas, including specific heat ratio, γ = 1.225, co-volume , b = 0.001 (m3 /kg) are obtained from Ref. [14]. mp
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number is 3200×800 points. The GBCM (Ghost Body-Cell Method) immersed boundary [15] is used to implemented the slip boundary on the solid wall. And the extrapolation and nonreflecting boundary was applied for the outer boundary.
3 Results and discussion 3.1 Code validation with experiments Fig. 3 The density distribution along the symmetric centerline at t = 0.0 (the projectile bottom arriving at the muzzle) with different background grids
For grid convergence tests, several uniform meshes with increasing resolutions have been used and the density distribution along the central axis at t = 0.0 are chosen as the demonstration (Fig. 3). When the calculated results of two meshes become small, the results are considered to be convergent. Our tests show that with the main background mesh resolution of 0.1905 mm × 0.1905 mm and the other moving mesh of 0.1905 mm ×0.09525 mm, the grid convergence can be reached. Due to the symmetry of the flow field, the computation was carried out only in the upper half of the computational domain. The background grids are spaced equally and the
To verify the mathematical model and numerical techniques in current study, two similar cases were chosen to compare. One is a projectile flying horizontally in the open air at a supersonic speed of 3.6 Ma, which is similar to present study but without the tube. The corresponding experimental shadowgraphs were taken by Charters [16] and are shown in Fig. 4a. Figure 4b is our corresponding numerical shadowgraph. It can be seen from the comparison that the agreement between the computational result and the experimental data is very good. However, due to the use of Euler equations for numerical simulation, the turbulent wake behind the cylinder cannot be visualized in Fig. 4b. To validate the GBCM immersed boundary condition used in our code, a plane incident shock wave with a Mach number of 2.4 passing through a square cylinder is chosen. Comparison of our computational shadowgraph with the experimental result obtained by Oertelal [16] is shown in Fig. 5. It can
Fig. 4 Comparison of experimental shadowgraph (a) and numerical shadowdiagram (b)
Fig. 5 Comparison of experimental (a) and numerical (b) shadowgraphs of shock-wave diffraction down a step
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Fig. 6 The computational shadowgraphs of muzzle flow field at t = 261.22 µs
be seen clearly that the curved diffracted wave and circular expansion wave from both Fig. 5a and b. Excellent agreement is observed for the curved diffracted wave and circular expansion wave, and the separation at the corner, which shows that the GBCM immersed boundary condition is acceptable in present study. 4 Results and discussion Figure 6 shows the computational shadowgraph of projectile out of the muzzle at t = 261.22 µs, and it is comparable with the corresponding experimental shadowgraph (Fig. 4) at almost the same time of Ref. [5] under the condition of ideal shoot and Fig. 1.2 of Ref. [17]. The jet core flow at the muzzle bounded by the Mach disk and barrel shock is shown clearly from Fig. 6. Our numerical simulation have shown that the muzzle blast flow induced by the relative high Mach number of projectile (Ma = 3.0) can also be identified as following four typical physical characters as described previously. The first can be partitioned as from the beginning of firing to the reach
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of the tube exit for the rear face of the projectile, and the flow consists mainly of precursor shock wave, contact surface, mach disc, etc., and can be featured as a typical supersonic jet flow. The second is similar to the first and is induced by the ejection of high pressure propellant gas behind the projectile, which generates the second jet flow. The third is connected with the moving of the projectile within the blast waves. The main process in this period includes the interaction between the projectile and the jet flows. The forth flow features with the projectile moving out of the precursor shock wave and occurrence of the bow shock wave. These prominent characteristics are discussed separately according to our predicted results. However, since the Euler model used for our simulation, therefore, some complex phenomena occurring in the third phase such as the shock-wake interaction around the projectile cannot be revealed in detail. Figure 7 shows two frames before the projectile driven out of the muzzle. To distinguish the contact surface and shock wave, both the isobars and the isopycnics are plotted at the lower and upper half of a single frame, respectively. The time, t, is set to zero when the bottom of projectile moves out of the muzzle. When the projectile accelerates inside the tube, the gas ahead of it is compressed, and results in the formation of the compression waves. When its speed is high enough, these waves rapidly coalesce into a steep shock front, i.e. precursor shock wave, which can be seen in Fig. 7a. When the projectile moves near to the muzzle, the precursor shock wave rushing out of the exit, its front changes from the planar to a spherical surface for the lateral expansion, and at the same time, the shock wave diffraction occurs at the exit. Following the precursor shock wave, the in-bore gas is driven out, and a typical Prandtl–Meyer expansion fan forms and spreads radially away from the muzzle (Fig. 7b), the fan-shaped expansion waves reflected from the jet boundary (shear layer) to form the barrel shock, which terminated in the vortex ring and the end of cap-like shock wave (Mach
Fig. 7 Density (upper half) and pressure (lower half) contours before the ejection of projectile
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Fig. 8 Density (upper half) and pressure (lower half) contours just after the ejection of projectile
Fig. 9 Density (upper half) and pressure (lower half) contours for projectile moving within the first blast wave
disc) as shown in Fig. 7b. The cap shock is spherical and its weaker rim sucked into a low-pressure core of the primary vortex ring. The contact discontinuity was formed between the in-bore gas and the air trapped by the precursor shock, i.e. air–air interface [17]. When the projectile is driven out of the tube, the high pressure propellant gas ( pe = 400.0 p0 ) behind it ejects out of the tube and gives rise to the second blast flow (Fig. 8a). Similar to the first blast, the leading shock wave and contact interface propagates radially along the axis. But the pressure and velocity are higher. Another difference is the background flow is still for the first blast, but for the second blast, the background flow is complicated due to the propagation and interaction of the first jet flow and projectile. Due to the existence of extremely low pressure and density behind the projectile, the leading shock wave and the contact interface of the second blast is invisible along the axis (Fig. 8b). At this time, the velocity of the leading shock wave is high compared to
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the projectile and the first precursor shock wave, therefore, it will catch up with the projectile and propagates forward along its side wall. When the projectile continues move forward, it will overtake the leading shock wave and penetrating through the contact interface, which leads to the generation of a bow shock wave in front of it (Fig. 9a). The diffraction of shock wave over the rear surface of the projectile which discussed in Ref. [6] are not be observed here, this maybe caused due to the narrow rear surface of the projectile. The second blast wave impacts and exceeds with the first contact surface, and intensifies the instable of the contact surface. Meanwhile, the first triple point between the second blast wave and the bow shock wave appears (Fig. 9b). Figure 10 shows that even though the second blast damps quickly, it will catch up with the first precursor shock wave due to its higher velocity. No obvious triple point occurs between the bow shock wave and the precursor shock wave
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Fig. 10 Density (upper half) and pressure (lower half) contours for projectile moving out of the first blast wave
Fig. 11 Distribution of Mach number (upper half) and density derivative (lower half) at t = 36.88 µs
as described in Ref. [6], since the time difference between the second blast wave overtaking the precursor shock wave and the projectile passing through the precursor shock wave is very small. This is caused by the higher Mach number of projectile resulting from the high initial ratio of pressure behind and ahead of the projectile in present study. Figure 11 shows the Mach number (upper half) and the density shadowgraph (lower half) distribution at t = 36.88 µs. Even after the projectile runs away from the exit, on the downstream of the muzzle, the typical Mach disk and barrel shock exist (Fig. 10), and they are wider and longer than usual ones. Outside of the barrel shock, the vortex ring become obvious and expands out at the corners of the Mach disk. The velocity of the projectile is almost steady after this period. The pressure distribution in front of the projectile along the center axis at different time is shown in Fig. 12. The origin of x-coordinate is defined at the exit of the tube. It can
Fig. 12 Pressure distribution in front of the projectile along the central axis
be seen that the precursor shock decays rapidly after the projectile ejected out of the tube from curve b to e. It is almost dissolved at curve f, t = 36.88 µs. The damp of the cap shock is also shown apparently in curve b, t = −22.79 µs. When the projectile overtake the cap shock into the subsonic flow area, the bow shock wave is formed in front of the projectile, and its intense increased with the projectile moving forwards and then become almost steady (curve d − f ).
5 Conclusions The launch process of a projectile from the tube is simulated numerically at a nearly realistic case. The Euler equations in the ALE forms and the dynamic chimera grid technique were applied to describe the high speed moving of the projectile and the AUSMDV scheme is employed to solve the equations. Compared to our previous relative low speed results
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calculated with Roe scheme, AUSMDV scheme can be used to simulate the higher Mach number of projectile launch. Our numerical results also show that even with the higher pressure ratio, the main muzzle blast field which features by four main processes significantly, but the interaction between the second blast and the projectile is more intensive, and the time for the projectile overtake the precursor shock wave is shorter. Acknowledgment The authors gratefully acknowledge the support from the Foundation of State Key Laboratory of Transient Physics.
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