DSMC investigation on flow characteristics of rarefied hypersonic flow over a cavity with different geometric shapes

International Journal of Mechanical Sciences 148 (2018) 496–509

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

DSMC investigation on flow characteristics of rarefied hypersonic flow over a cavity with different geometric shapes Guangming Guo a,∗, Qin Luo b a b

College of Mechanical Engineering, Yangzhou University, Yangzhou 225127, China College of Information Engineering, Yangzhou University, Yangzhou 225127, China

a r t i c l e

i n f o

a b s t r a c t

Keywords: Cavity Flow characteristics Shear layer Recirculation region DSMC

As one of the most fundamental configurations, the cavities with different geometric shapes are frequently encountered on various aerodynamic surfaces. In this paper, the direct simulation Monte Carlo (DSMC) which is one of the most successful particle simulation methods in treating rarefied gas dynamics is employed to investigate the flow characteristics of the cavity with a length-to-depth ratio of 1–8, a rearwall-to-frontwall height ratio of 0.5–2, and an inclined frontwall ranging from 20° to 70° for the free stream at a Mach number of 8 and an altitude of 60 km. The simulation results indicate that the shear layer and recirculation region within the cavity are changed greatly by varying length-to-depth ratio and the cavity is transformed into a closed type at the length-to-depth ratio of 6. For the cavity with a fixed length-to-depth ratio, the increase in rearwall height would push the shear layer away from cavity floor and hence increases the area of recirculation region. Conversely, the decrease in rearwall height would draw the shear layer into the interior of the cavity and finally the recirculation region would be split into two smaller and separated ones located at the left and right corners of the cavity respectively. A significant finding is that the criterion (i.e., length-to-depth ratio) for classifying the cavity type is not a constant but changes with freestream conditions. In addition, recirculation region(s) within the cavity can be eliminated completely by simultaneously inclining the frontwall and rearwall with an appropriate angle and meanwhile the shear layer is affected slightly.

1. Introduction

and triangular cavities have larger turbulence intensity compared with semicircular cavity. Huang [10] investigated the influence of cavity configuration on the combustion flow field performance of the hypersonic vehicle, and he found that the cavity with a sweptback angle of 45° could further improve the aero-propulsive performance of the hypersonic vehicle. He also investigated the influence of geometric parameters on the drag force of the heated flow field around the cavity by variance analysis method [6]. Luo [5] estimated and compared the drag forces of three different geometric shapes of the cavity flame holder, namely the classical rectangular, triangular and semi-circular, and the cavities are with the fixed depth and length-to-depth ratio. He found that the triangular cavity imposes the most additional drag force on the scramjet engine, and the drag force of the classical rectangular one is the least. There are two basic flow structures formed by the free stream passing through the cavity, namely the shear layer and low-speed recirculation region within the cavity. According to flow characteristics of the shear layer, the cavities can be classified as open and closed types [11]. For an open cavity, usually with length-to-depth ratio (L/D) less than 10, the shear layer spans the whole cavity and a large recirculation region formed underneath the shear layer. While for a closed cavity, which is

The cavity is one of the most fundamental configurations and the cavities with different geometric shapes are frequently encountered on various aerodynamic surfaces, such as spacecraft hulls, gas-turbine channels, and surfaces with ribbing in heat exchangers and microelectronic chips [1]. Recently, the wall-mounted cavity has been widely employed in the scramjet engine to prolong the residence time of the supersonic air and the injected fuel in the combustor [2]. In addition, optical seeker of some missiles usually has the cavity-shaped optical window for light beam transmission. Therefore, the cavities with varying shapes have been investigated numerically and experimentally by many researchers [3–9]. Vinha studied the incompressible fluid flow over a rectangular open cavity for understanding evolution of the span-wise instabilities of flow and interactions between different dynamic modes with the dynamic mode decomposition algorithm [3]. The characteristics of the cavities with different shapes in the free stream with low velocity were experimentally tested by Ozalp [4], he observed that the maximum turbulence intensities occur at cavity lid in the centerline section and rectangular

∗

Corresponding author. E-mail address: [email protected] (G. Guo).

https://doi.org/10.1016/j.ijmecsci.2018.09.022 Received 4 May 2018; Received in revised form 6 September 2018; Accepted 14 September 2018 Available online 15 September 2018 0020-7403/© 2018 Published by Elsevier Ltd.

G. Guo, Q. Luo

International Journal of Mechanical Sciences 148 (2018) 496–509

typically long and shallow with an L/D greater than 10, the shear layer would impinge on the cavity floor, leads to two smaller recirculation regions located at the left and right corners of the cavity respectively. Rowley et al. [12] employed two-dimensional direct numerical simulations (DNS) to study subsonic flows over rectangular cavities. The results revealed that the shear layer would become absolutely instable for longer cavities and higher Mach numbers. Das et al. [13] experimentally investigated the influence of cavity width on the shear layer oscillations, the results shown that the cavity length-to-width ratio plays a dominant role in determining the shear layer oscillations. Recently, Wang et al. [14] investigated entrainment characteristics of the shear layer in supersonic flows using two-dimensional large eddy simulations (LES), it is found that entrainment of the free stream into the recirculation region is mainly determined by the interactions between the shear layer and the cavity rearwall. For growth rate of the shear layer, it decreases with increasing Mach numbers [15,16], which seems like the compressibility effects observed in mixing layers. Near space is generally defined as the zone between 20 and 100 km of the atmosphere, including a majority of regions in the stratosphere, the mesosphere and partial regions in the thermosphere, and it has been paid more and more attentions in recent years due to its military application value [17]. The flying vehicles in near space usually have hypersonic speeds (i.e., M∞ ≥ 5), namely the hypersonic vehicles. For example, the hypersonic cruise vehicle concept in the Falcon program has a cruise speed approaching Mach number 10 [18]. Kumar [19] analyzed aero-thermal characteristics of a hypersonic vehicle with two lifting body configurations at Mach number 7, and described the aero-thermal environment for several critical regions of the hypersonic vehicle, such as nose-cap, leading edges, fuselage-upper surface, and fuselage-lower surface. Gerdroodbary [20] numerically simulated a hypersonic flow (M∞ = 5.75) over highly blunted cones with spike at different angles of attack, and the numerical results presented the pressure distributions and the surface heat reduction for different type of spike. Flow parameters and skin temperature distributions caused due to aerodynamic heating were simulated and analyzed by Murty for a high speed (∼5Ma) aerospace vehicle at about 25 km altitude using commercial CFD code [21]. Mohammadzadeh et al. [22] performed a DSMC investigation to study the nonequilibrium effects on monatomic and diatomic rarefied flows in micro/nano lid-driven cavity and investigated thermal characteristics of micro- or nanocavity flow [23], they observed and confirmed the unconventional cold-to-hot heat transfer, and attributed such phenomena to the sharp bends in the velocity profiles which take place near the top corners of the cavity. Palharini et al. [24] carried out a computational investigation of rarefied hypersonic gas flows in the transitional flow regime over 3D cavities by DSMC method and they focused on the flowfield structure characterization under a rarefied environment and in the presence of chemical reactions. Recently, Roohi et al. [25] investigated rarefied gas flows in nano-scale isosceles triangular cavities with a motion away from the square corner using DSMC method and the results show that vortical characteristics of the flow inside triangles are influenced not only by the geometrical configuration but also by the rarefaction effects. However, to the best of the authors’ knowledge, the investigation on flow characteristics of rarefied hypersonic flow over a cavity, especially for the influence of geometric shapes of the cavity on the shear layer and recirculation region behaviors, has not appeared in the open literature. In addition, further understanding the flow characteristics of a cavity with different geometric shapes in rarefied hypersonic flows is extremely necessary for optimum design of hypersonic vehicles because the cavity configuration widely exists in some key parts of them. Therefore, the lack of such results motivates the current study. The objective of this work is to investigate the flow characteristics of rarefied hypersonic flow over a cavity with different geometric shapes using direct simulation Monte Carlo (DSMC) method, which is currently the only available tool for the solution of rarefied flow fields from free

molecular to continuum low density regimes [22,23,26,27], and this is attributed to the fact that the DSMC is a particle based microscopic method that converges to the solution of the Boltzmann equation in the limit of infinite simulating particles and the Boltzmann equation is able to describe the flow behaviors of gas at every rarefaction degree. The rest of this paper is organized as follows. The DSMC method is introduced and validated in Section 2. In Section 3, simulation cases and grid independence analysis are presented. Section 4 presents several evaluation parameters for flow characteristics analysis. The results of the considered cases and discussion are given in Section 5. Finally, the significant conclusions are presented in Section 6. 2. Numerical method and code validation 2.1. DSMC introduction The air density decreases gradually with increasing altitude, and thus the rarefied gas effects become serious in near space where the flow varies from free-molecular to near-continuum low density regimes. As the rarefaction degree increases, the theoretical assumption in the conventional constitutive relations for continuous flow, such as the NavierStokes equation, loses its validity [28], whereas the Boltzmann equation is able to describe the behavior of a gas flow at every rarefaction degree [29]. The Boltzmann equation is given as [30,31]: 𝜕𝑓 1 + 𝜐 ⋅ ∇𝑥 𝑓 = 𝑄 (𝑓 , 𝑓 ), (1) 𝑥 ∈ ℝ𝑑𝑥 , 𝜐 ∈ ℝ𝑑𝜐 . 𝜕𝑡 𝐾𝑛 Where 𝑓 (𝑡, 𝑥, 𝜐)is the density distribution function of a dilute gas at positionx, with velocity𝜐and at timet, Kn is the Knudsen number, and the bilinear collision operator Q(f, f)describes the binary collisions of the particles and is defined by 𝑄(𝑓 , 𝑓 )(𝜐)=

∫

∫

)[ ( ) ( ) ( )] ( 𝜎 |𝜐 − 𝜐1 |, 𝜔 𝑓 𝜐′ 𝑓 𝜐′ ∗ − 𝑓 (𝜐)𝑓 𝜐∗ 𝑑 𝜔𝑑 𝜐∗

(2)

ℝ𝑑𝜐 𝕊𝑑𝜐−1

where𝜔is a unit vector on the sphere 𝕊𝑑𝜐−1 which is the unit sphere defined inℝ𝑑𝜐 space, the velocity (𝜐′ , 𝜐′ ∗ )represents the post-collisional velocities whose relation to the pre-collisional velocities (𝜐′ , 𝜐′ ∗ ) are given by ) ( ) 1( 𝜐′ = (3) 𝜐 + 𝜐∗ + ||𝜐 − 𝜐∗ ||𝜔 , 𝜐′ ∗ = 12 𝜐 + 𝜐∗ − ||𝜐 − 𝜐∗ ||𝜔 . 2 In Eq. (2), 𝜎is the nonnegative collision kernel which depends on the model of forces between particles, and we also define the total cross section 𝜎 T as ( ) 𝜎𝑇 ||𝜐 − 𝜐∗ || = ∫

) ( 𝜎 ||𝜐 − 𝜐∗ ||, 𝜔 𝑑𝜔.

(4)

𝕊𝑑𝜐−1

In the case of inverse kth power force between particles, it has the form ( ) 𝛼 𝜎 ||𝜐 − 𝜐∗ ||, 𝜃 = 𝑏𝛼 (𝜃)||𝜐 − 𝜐∗ || , (5) where 𝜃 is the collision angle of simulating particles, a is a viscositydependent constant, ba (𝜃) is a function about 𝜃 and its specific expression is related to the particle collision model. Withf, the macroscopic density𝜌, mean velocityu, and temperatureT, can be obtained as 𝜌=

∫

ℝ𝑑𝜐

𝑓 𝑑𝜐,

𝑢=

1 𝜐𝑓 𝑑𝜐, 𝜌∫ ℝ𝑑𝜐

𝑇 =

1 𝑑𝜐𝜌

∫

|𝜐 − 𝑢|2 𝑓 𝑑𝜐

(6)

ℝ𝑑𝜐

In addition, the collision operator (i.e., Eq. (2)) satisfies the following two important properties: •

Conservation laws: ∫

𝑄(𝑓 , 𝑓 )𝜙(𝜐)𝑑𝜐 = 0,

𝑓 𝑜𝑟

𝜙(𝜐) = 1, 𝜐, |𝜐|2 ,

ℝ𝑑𝜐

which gives conservation of mass, momentum and total energy. 497

(7)

G. Guo, Q. Luo •

International Journal of Mechanical Sciences 148 (2018) 496–509

Boltzmann’s H theorem: 𝑑 𝑓 log 𝑓 𝑑𝜐 = 𝑄(𝑓 , 𝑓 ) log 𝑓 𝑑𝜐 ≤ 0, ∫ 𝑑𝑡 ∫

(8)

which implies that any system reaches its equilibrium state at which the entropy − ∫ 𝑓 log 𝑓 𝑑𝜐 is maximum. The equilibrium distribution function has the form of a local Maxwellian distribution: M(𝜌, 𝑢, 𝑇 )(𝜐) =

𝜌 (2𝜋𝑇 )𝑑𝜐∕2

( ) |𝑢 − 𝜐|2 exp − . 2𝑇

(9)

The Boltzmann equation can be solved by stochastic schemes, commonly known as DSMC. The DSMC method, first proposed by Bird, is a particle based microscopic method that converges to the solution of Boltzmann equation in the limit of infinite simulating particles [26]. In this method, simulating particles represent a cloud of gas molecules that travel and collide with each other and solid surfaces, and the interactions with surfaces and with other molecules conserve both momentum and energy. The primary principle of DSMC is to decouple the motion and collision of particles during one time step, so the computational time step should be smaller than the physical collision time. The present DSMC computations used the variable hard sphere (VHS) model to simulate particles collision and the Larsen-Borgnakke model acts as the internal energy model [26]. In every time step, all simulating particles move according to their individual velocities and in each cell, a certain number of collision pairs are selected using the no-time-counter (NTC) method and then collisions are calculated. When the flow field is steady, the macroscopic properties of the gas, such as velocity, temperature, density, shear stress, pressure and so on are obtained by taking appropriate sampling of microscopic properties of the simulating particles. The micro-perspective and statistical behavior are the distinct features of DSMC, and working process of the DSMC can be summarized as the following steps [31]: (a) Read grid data and record the information of the boundary condition; (b) Initialize flow field and calculate the entering number of simulated molecules; (c) Simulated molecules move and interact with boundary; (d) Index all of the simulated molecules; (e) Simulated molecules are probabilistically selected and collide each other; (f) Sample the mesh cell and wall information and repeat steps (c)–(f) until reaching steady state; (g) Write out information of the flow field and wall. Over the past few decades, the DSMC method has been the predominant predictive tool in rarefied gas flows. However, computational consumption is probably the main blockage factor existed in the extensive application of DSMC. Generally speaking, there are two different ways to solve this difficulty when facing massive computation. One way is to improve the DSMC method, such as with the MPC algorithm [32], or the AP (Asymptotic-Preserving) algorithm [30]. The other is to introduce some assistant techniques, such as an efficient parallel technology [33]. In our current DSMC code, the parallel technique is employed. The following numerical simulations are performed using our parallel two-dimensional DSMC code. According to our experience in DSMC computations, the time step is set to 0.05 μs which is smaller than the mean molecule collision time, the times of sampling after a steady flow are set to be 20,000 for a sufficiently small statistical error, and we use 30 particles per mesh cell to reduce the influence of statistical dependence between model particles. Because DSMC is essentially a statistics method [34], it should be noted that the macroscopic flow properties of the cavity flow field, such as velocity, temperature, density, shear stress, pressure and so on are obtained by taking appropriate sampling of microscopic properties of the simulating particles after achieving steady flow. In other words, the following presented DSMC results of all cases we considered actually reflect the time-averaged flow properties of rarefied hypersonic flow over a cavity with different geometric shapes.

Fig. 1. Geometry and dimensions of the biconic structure for validating the DSMC. Table 1 Freestream conditions for the biconic problem. Gas

V∞ (m/s)

Re∞ (m−1 )

P∞ (pa)

T∞ (K)

Ma∞

𝜌∞ (kg/m3 )

N2

2073

1.37467e5

2.23

42.6

15.6

1.757e-4

2.2. Validation of the DSMC code As a matter of fact, the DSMC code used in this paper have been validated with some typical benchmark cases in our recent works [30,31,33,17,35] in which the validity of the DSMC code was checked completely, and the detailed information can be found in those documents. In this paper, the DSMC code is run for a biconic problem which has available experimental data about some parameters over its surface for validating the current DSMC quantitatively. The biconic geometry is presented in Fig. 1 [36], and the freestream conditions for this biconic problem are listed in Table 1 [37]. Due to axial symmetry of the biconic structure, only half of its geometric region is used to compute the hypersonic flow over the biconic structure. The obtained DSMC results and the comparison between numerical and experimental surface heat flux and pressure are shown in Fig. 2. It is seen that there is a recirculation zone over the connection of two cones, forming a separation point and a reattachment point before and after it respectively, which agree well with the numerical results in Refs. [36,37]. As displayed in (c) and (d) of Fig. 2, both the calculated surface heat flux and pressure distribution over the biconic agree reasonably well with the experiment. Therefore, the comparisons indicate that the employed DSMC code has acceptable accuracy in simulating rarefied hypersonic flow. 3. Simulation cases and grid independency analysis 3.1. Simulation cases As mentioned above, the cavity shape crucially affects aerodynamic behaviors of the flow structures inside the cavity, so different geometric shapes for a cavity is necessary to consider to reveal the specific flow 498

G. Guo, Q. Luo

International Journal of Mechanical Sciences 148 (2018) 496–509

Fig. 2. Contours of (a) pressure and (b) temperature surrounding the connection of two cones and comparison of (c) surface heat flux and (d) pressure distribution over the biconic. Table 2 Hypersonic free stream conditions for the current study.

Hypersonic free stream

P∞ (pa)

T∞ (K)

Ma∞

f(1/s)

Lm (m)

𝜌∞ (kg/m3 )

21.958

247.0

8.0

1.62e6

2.62e-4

3.09e-4

Note: Data in Table 2 come from U.S. Standard Atmosphere, 1976.

characteristics, and this is the main focus of the current work. Three cavity configurations, namely the cavity with a length-to-depth ratio of 1–8, a rearwall-to-frontwall height ratio of 0.5–2 and an inclined frontwall ranging from 20° to 70°, respectively, as shown in Fig. 3, are taken into account and the corresponding simulation cases are designated as B1–B8, R1–R3, and F1–F4. The geometric parameters of these cavities are as follows: the depth (D), the length (L), the rearwall height (Da) and the frontwall inclination angle (𝜃). The depth of the cavity is set to be a constant in this paper, namely D = 10 mm, and other geometric parameters are all normalized with D. The origin of coordinate system is located at the lower left corner of the cavity, x and y denote the streamwise and vertical directions, respectively. To obtain rarefied hypersonic flow for this study, the free stream conditions corresponding to an altitude of 60 km are chosen and the free stream Mach number is set to be 8.0 for every simulation case. The free stream pressure (P∞ ), temperature (T∞ ), density (𝜌∞ ), molecule collision frequency (f) and mean free path (Lm ) at the altitude of 60 km are listed in Table 2. As shown in Fig. 3, the free stream conditions are imposed on the left boundary of the computational zone, the right boundary is an outflow condition, the upper boundary is a free interface and the remaining boundaries are walls. The free stream consists of 78% N2 and 22% O2

and its molecular weight is about 28.96 kg/mol. In addition, impenetrable and constant temperature conditions are applied for walls. 3.2. Grid independence analysis From the viewpoint of mathematics, the DSMC method is physically based on the Boltzmann equation, which simulates the same physics as the Boltzmann equation by using a large number of model particles to emulate the real gas flow, instead of providing a direct solution of the Boltzmann equation. That is, the DSMC is essentially a molecule method and macroscopic properties of the gas (e.g., velocity, temperature, density, shear stress, pressure, etc.) are obtained by taking appropriate sampling of microscopic properties of the simulating particles after a steady flow, so the accumulation of errors does not exist in DSMC simulations. In fact, the calculation precision of DSMC method is mainly determined by grid density, times of sampling, particles in each mesh cell and time step. As mentioned in Section 2.1, the times of sampling, particles in each mesh cell and time step are set to adequate values according to our experience in DSMC computations, so just the grid density is discussed in this subsection. The B6 case is taken as an example to test the grid independence with three grid scales: coarse grid (86,000 mesh cells), moderate grid 499

G. Guo, Q. Luo

International Journal of Mechanical Sciences 148 (2018) 496–509

Fig. 3. Schematic of the three cavity configurations employed in the current study along with their corresponding simulation cases.

The three grid scales are successively employed to perform DSMC computations for the B6 case, and then the streamwise velocity (𝑈 ) and temperature (T) profiles along vertical line of x/D = 4 are extracted from computational data and are shown in Fig. 5. It is seen that both streamwise velocity and temperature profiles of every grid scale are agree well with each other and the enlarged views do not show any significant discrepancy for the three grid scales, especially for the moderate and refined grids. For this reason, the moderate grid scale is proportionally applied to all the simulation cases in the following DSMC computations in order to reduce computation consumption. Fig. 4. Computational grid for the B6 case.

4. Evaluation parameters of the flow characteristics Before the computed results are analyzed, it is necessary to present several evaluation parameters for describing the flow characteristics quantitatively. Fig. 6 is a schematic of the free stream passing through an open cavity, in which the resulting flow structures are shown clearly, namely the upstream boundary layer, the shear layer spanning the cavity, the large recirculation region underneath the shear layer, and the downstream boundary layer. Obviously, the shear layer and recirculation region are the main flow structures for the flow over a cavity. For the shear layer, it is usually characterized by the growth rate, which is

(210,000 mesh cells) and refined grid (323,000 mesh cells). In this paper, all of the 2D grids are generated using the available commercial software POINTWISE. As shown in Fig. 4, the grid is densely clustered near walls, inside the cavity, in vicinity of the step edge and in the shear layer zone for the purpose of acquiring flow structures accurately. It should be noted that most of the grid points in Fig. 4 are actually skipped for the purpose of clarity. 500

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International Journal of Mechanical Sciences 148 (2018) 496–509

Fig. 5. Comparison of streamwise velocity (left) and temperature (right) profiles along the vertical line of x/D = 4 for the three grid scales to test the grid independence.

Fig. 6. Schematic of flow structures produced by free stream passing through an open cavity. Fig. 7. The left and right recirculation regions of the B7 case that revealed by streamlines.

equivalent to the shear layer thickness throughout the length of the cavity. For the shear layer over the cavity, it is actually formed by flow separation of the freestream passing through the upstream step, which also brings a recirculation region below the shear layer and inside the cavity (see Fig. 6). Consequently, the upper boundary flow velocity of the shear layer is close to the free stream velocity while the lower boundary flow velocity of the shear layer is about equal to the rotational velocity of the recirculation region. That is, both the upper and lower boundaries of the shear layer can be defined according to flow velocity. Unfortunately, however, there is so far not an authoritative definition of boundary of the shear layer over the cavity. In this study, the upper and lower boundaries of the shear layer are defined as 90% and 10% isolines of the free stream velocity, respectively. The shear layer thickness at any one streamwise position is then calculated using Eq. (13). Note that the maximum of shear layer thicknesses throughout the length of the cavity, denoted by 𝛿 s , and thickness of the downstream boundary layer at the outlet, denoted by 𝛿 b , are used as the evaluation parameters to describe the influence of different cavity shapes on the shear layer and downstream boundary layer. 𝛿(𝑥) = 𝑦𝑈 ∕𝑈

𝑓 𝑟 =0.9

− 𝑦𝑈 ∕𝑈

𝑓 𝑟 =0.1

Fig. 8. The illustration of obtaining positions of reattachment points of the left and right recirculation regions for the B7 case.

(13)

In Eq. (13), the Ufr denotes streamwise velocity component of free stream and 𝑈̄ denotes time-averaged streamwise velocity. For the recirculation region, two evaluation parameters, namely streamwise length and inclination angle, are used to describe size and posture of the recirculation region. Fig. 7 shows the recirculation regions of the B7 case, where xR,left and xR,right are the streamwise lengths of the left and right recirculation regions respectively, which are defined as the streamwise distance between the reattachment point (i.e., xR 1 and xR 2 ) and the fronwall/rearwall. There are some different ways to obtain the position of the reattachment point, for example, it may be defined as the loca-

tion where the time-averaged skin friction coefficient is zero [38] or defined as the location where the time-averaged streamwise velocity is zero [31], and the latter is employed in this paper to obtain the positions of reattachment points to calculate streamwise lengths of the left and right recirculation regions, as illustrated by Fig. 8. Recently, we proposed a new parameter, i.e. inclination angle, to describe the posture of the recirculation region for the backward-facing step configuration [31] and it is also used for the current cavity con501

G. Guo, Q. Luo

International Journal of Mechanical Sciences 148 (2018) 496–509

In order to quantitatively analyze variation of the shear layer and downstream boundary layer with increasing L/D, borderlines of shear layer and boundary layer for B1–B8 cases are extracted from Fig. 10 and shown in Fig. 11, where the 𝛿 s and 𝛿 b of every case are marked. There are two features concerning the shear layer and boundary layer. First, the boundary layer thickness approximately increases linearly with the streamwise distance and the upstream boundary layer is not affected by the varying L/D ratios. Second, the shear layer thickness variation is irregular and both 𝛿 s and 𝛿 b appear to increase with L/D. The detailed information about 𝛿 s and 𝛿 b are given in Fig. 12. As the L/D ratio increases, it is seen that both 𝛿 s and 𝛿 b increase with an approximately linear rate for L/D ⟨ 6. However, the effect of different L/D ratio on 𝛿 s and 𝛿 b becomes negligible when L/D ⟩ 6. Also, it is observed that the 𝛿 s is always larger than 𝛿 b , especially for L/D > 6. For example, the 𝛿 s is about 33% larger than 𝛿 b at L/D = 8. The recirculation regions of B1–B8 cases are extracted from Fig. 10 and shown in Fig. 13. At the same time, the streamwise length and inclination angle of every recirculation region are calculated according to their definitions illustrated in Section 4 and the obtained results are also presented in Fig. 13 to reveal evolution of the recirculation region with increasing L/D ratio from a mathematical point of view. It is seen that, for B1–B5 cases, namely the L/D < 6, there is only one recirculation region occupying nearly the whole area of the cavity and thus the streamwise length (xR ) of the recirculation region is actually equal to the cavity length L. With regard to the B6 case, it is slightly knotty to determine how many recirculation regions within the cavity because there are two eddies inside the cavity, like B7 and B8 cases, but meanwhile the two eddies are still connected by the fluid near the cavity floor. However, given that the recirculation region is essentially produced by the shear layer and the situation of B6 case is just the continuation of B5 case, so we think there is only one recirculation region for B6 case. As the L/D ratio further increases (see B7 and B8), two smaller recirculation regions clearly appear inside the cavity, but the streamwise length of every recirculation region decreases obviously due to serious extrusion effect from the expanded shear layer, especially for the one at the left corner of the cavity. It is also found that the streamwise length of every recirculation region of B8 case is slightly larger than that of B7 case. Fig. 14 shows inclination angle variation of the recirculation region with increasing L/D ratio, where 𝜑 denotes the inclination angle for B1– B5 cases which have only one recirculation region inside the cavity, the 𝜑left and 𝜑right denote the inclination angles of the left and right recirculation regions for B6–B8 cases, respectively. As shown in Fig. 14, where the inclination angle of recirculation region for the cases of L/D = 10, 13, and 16 are also given together to see how the inclination angle of recirculation region changes with the L/D. It is found that the inclination angle keeps a constant of 90° for L/D < = 4, then, it begins to decrease with the further rise in L/D until L/D = 7, at which the 𝜑left has a value of about 45° and the 𝜑right has a value of about 70° When the L/D is larger than 7, both the 𝜑left and 𝜑right change slightly with L/D increases. For example, the difference of 𝜑left and 𝜑right for L/D ratios ranging from 7 to 16 are less than 5°. In other words, for the closed cavity, shape of its recirculation regions is approximately invariable as the L/D increases. According to the analysis above, L/D = 6 is a critical value for the baseline cavities, such as the cavity is an open type when L/D < 6 while it is a closed type when L/D > 6 (see Fig. 10). For this reason, the cavity with L/D = 6 is chosen as the reference cavity to further investigate the influence of varying Da/D and 𝜃 on the flow characteristics, and the details are presented in the following Sections 5.2 and 5.3, respectively.

Fig. 9. Schematic of the inclination angles of the left and right recirculation regions.

figuration in the same way. The definition of inclination angles of the left and right recirculation regions is illustrated by Fig. 9, where points 1 and 2 are both in the external streamline of the recirculation region and meanwhile points 1 and 2 have maximums in y and x directions, respectively. The 𝜑left and 𝜑right denote the inclination angles of the left and right recirculation regions, respectively. So far, four evaluation parameters, namely maximum thickness of the shear layer (𝛿 s ), downstream boundary layer thickness at the outlet (𝛿 b ), streamwise length (xR ) and inclination angle (𝜑) of the recirculation region, have been introduced and they are used to describe the flow characteristics for all the simulation cases. 5. Results and discussion In this section, we present the DSMC simulation results of rarefied hypersonic flow over a cavity with different geometric shapes. As shown in Fig. 3, the geometrical parameters that changed are the cavity lengthto-depth ratio (L/D), rearwall-to-frontwall height ratio (Da/D) and frontwall inclination angle (𝜃), which are used to find out the influence of varying L/D, Da/D, and 𝜃 on the flow characteristics. Because the DSMC is essentially a statistics method, the computational results presented in this section reflect the time-averaged properties of the shear layer and recirculation region in every case. 5.1. Flow characteristics of the cavity with different length-to-depth ratios The Mach number contours for B1–B8 cases are shown in Fig. 10, where the borderlines of boundary layer and shear layer along with the recirculation region revealed by streamlines are also presented. It can be observed that, when the length-to-depth ratio (L/D) is gradually increasing from L/D = 1.0 to L/D = 8.0, the shear layer undergoes a significant change, namely it spans the whole cavity for the cases of L/D smaller than 6, whereas it begins to impinge on the cavity floor when L/D larger than 6. As a result, the resulting recirculation region, which is almost a circular shape in B1 case, gradually becomes a long and narrow shape with the increasing L/D due to the extrusion effect from shear layer, and finally it is split into two smaller recirculation regions located at the left and right corners of the cavity respectively. The length-to-depth ratio is usually used to classify the cavities and the cavity is an open type for L/D ⟨ 10 while it is a closed type for L/D ⟩ 10 [6]. According to the current calculation results, however, the cavity has already become the closed type when L/D > 6 (see B7 and B8). That is to say, the previous criterion for classifying the cavity type is probably no longer practicable for the cases of rarefied hypersonic flow over a cavity. In Ref [24], the cavity depth was kept constant at 3 mm and the cavity length assumed different values ranging from 3 to 15 mm (i.e., L/D = 1, 2, 3, 4, and 5), and the freestream conditions used corresponded to that experienced by a re-entry vehicle at a velocity of 7600 m/s and an altitude of 80 km. At this condition, the cavity type becomes to transform even for L/D = 3, and it is the closed cavity for L/D = 4 and 5. Therefore, a significant finding is that the criterion (i.e., L/D) for classifying the cavity type is not a constant but changes with freestream conditions. According to our currently computed results (see Fig. 10) and those in Ref [24], it can conclude that the criterion (i.e., L/D) would become smaller for the freestream with a higher altitude and a larger velocity.

5.2. Flow characteristics of the cavity with different Da/D ratios In order to analyze the influence of different Da/D ratios on the flow characteristics of rarefied hypersonic flow over the cavity, the cases of the rearwall higher and lower than the frontwall are all taken into account, namely the Da/D ratio changes from 0.5 to 2. The Mach number 502

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Fig. 10. Mach number contours, borderlines of boundary layer and shear layer along with the recirculation regions for B1–B8 cases.

Fig. 11. The boundary layer and shear layer for B1–B8 cases, the two isolines denote U /Ufr = 0.1 and 0.9, respectively.

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Fig. 12. Variation of 𝛿 s and 𝛿 b with increasing L/D ratio.

Fig. 14. Inclination angle variation of the recirculation region(s) with increasing L/D ratio.

contours for R1–R3 and B6 cases are shown in Fig. 15, where the borderlines of boundary layer and shear layer along with the recirculation region revealed by streamlines are also presented. Note that the B6 case reproduced here is used to contrast with the R1–R3 cases. As shown in Fig. 15, the Da/D ratio has a significant effect on both shear layer and recirculation region. Specifically, for the case of the rearwall lower than the frontewall (see R1), the shear layer is more close to the cavity floor due to the decrease in rearwall height, results in two smaller and separated recirculation regions within the cavity. For the cases of the rearwall higher than the frontewall (see R2 and R3), the shear layer moves upward and rapidly away from the cavity floor due to the increase in rearwall height, which greatly increases the area of the recirculation region and finally produces a large recirculation region occupying the whole cavity. It is also observed that the higher rearwall, the larger recirculation region. On the contrary, the shear layer becomes thinner as the rearwall rises.

The borderlines of boundary layer and shear layer for R1–R3 and B6 cases are extracted from Fig. 15 to reveal the influence of rearwall height on the shear layer and downstream boundary layer, as shown in Fig. 16, where the 𝛿 s and 𝛿 b of every case are marked. Obviously, as the Da/D ratio increases, the upstream boundary layer is not affected while both 𝛿 s and 𝛿 b decrease gradually because the risen rearwall acts as a compression ramp for the shear layer, imposing compression effects on the shear layer. The specific values of 𝛿 s and 𝛿 b for different Da/D ratios are given in Fig. 17. Specifically, the 𝛿 s at Da/D = 2 is about 22% lower than that at Da/D = 0.5, and the 𝛿 b at Da/D = 2 is about 36% lower than that at Da/D = 0.5. In addition, it is found that the 𝛿 s is always larger than 𝛿 b when the Da/D ratio varies from 0.5 to 2. The recirculation regions of R1–R3 and B6 cases that extracted from Fig. 15 along with the calculated streamwise length and inclination an-

Fig. 13. Evolution of the recirculation region with increasing L/D ratio. 504

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Fig. 15. Mach number contours, borderlines of boundary layer and shear layer along with the recirculation regions for R1–R3 and B6 cases.

Fig. 16. The boundary layer and shear layer for R1–R3 and B6 cases, the two isolines denote U /Ufr = 0.1 and 0.9, respectively.

increasing Da/D ratio. From the above analysis, a significant conclusion is that the cavity type can also be transformed by adjusting the rearwall height for the cavity with a fixed L/D. For example, the B6 cavity becomes an open type when its rearwall height is increased (see R2 and R3 cases) while it converts to a closed type when its rearwall height is decreased (see R1 case), as shown in Fig. 15. With the increasing of the Da/D ratio, as shown in Fig. 19, the inclination angles of both left and right recirculation regions gradually increase for Da/D < = 1.5, at which the inclination angle has a maximum of about 87°. Then, with the further increase in the Da/D ratio, the inclination angle begins to decrease due to the increase in rearwall height (see R2 and R3 cases in Fig. 18).

Fig. 17. Variation of 𝛿 s and 𝛿 b with increasing Da/D ratio.

5.3. Flow characteristics of the cavity with an inclined frontwall The Mach number contours, borderlines of boundary layer and shear layer along with the recirculation regions for F1–F4 cases are shown in Fig. 20 to reveal the influence of frontwall inclination on flow characteristics. It is seen that, there is no recirculation region in the left corner of the cavity when the frontwall inclination angle 𝜃 is 20°, 30°, and 50°, whereas a very small recirculation region appears in the left bottom of the cavity for 𝜃 = 70°. As the increase of 𝜃, the shear layer away from the cavity floor slowly, leads to a little increase in the area of low-speed flow inside the cavity. At the same time, the thickness of shear layer and downstream boundary layer increases slightly with the increase of 𝜃, while the right recirculation region seems to be not affected by the variation of 𝜃.

gle of every recirculation region are all presented in Fig. 18, where the evolution of the recirculation region with increasing Da/D ratio is revealed clearly. Comparing with the B6 case, when the rearwall height decreases (see R1), two recirculation regions appear inside the cavity associated with a quick reduction in streamwise length of the right recirculation region. In this case, the R1 cavity actually belongs to a closed type. Conversely, when the rearwall height increases (see R2 and R3), it pushes the shear layer upward and thus increases the area of the recirculation region, which leads to a quite large recirculation region formed within the cavity, as a open cavity does (see B1–B5 cases in Fig. 13). In addition, the recirculation region thickness increases gradually with the 505

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Fig. 18. Evolution of the recirculation region with increasing Da/D ratio.

but their increments are quite small for the 𝜃 changing from 30° to 90° Specifically, the 𝛿 s at 𝜃 = 90° is about 0.16D larger than that at 𝜃 = 30° and the 𝛿 b at 𝜃 = 90° is only 0.09D larger than that at 𝜃 = 30° In addition, it is observed that the 𝛿 s is always larger than 𝛿 b when the 𝜃 varies from 30° to 90° The recirculation regions of F1–F4 cases that extracted from Fig. 20 along with the calculated streamwise length and inclination angle of every recirculation region are all given in Fig. 23, where the evolution of the recirculation region with increasing 𝜃 is revealed clearly. It is seen that there is always a large recirculation region located at the right side of the cavity for 𝜃 = 20°, 30°, and 50°. When the 𝜃 increases to 70°, a very small recirculation region with streamwise length (xR , left ) of 0.62D and inclination angle of about 50° appears at the left bottom of the cavity. For the right recirculation region which is the dominated flow structure within the cavity for F1–F4 cases, its posture and streamwise length change slightly with the increasing 𝜃, as revealed in Fig. 24. It is observed that the increment of streamwise length (xR , left ) of the right recirculation region is about 0.13D for 𝜃 varying from 30° to 90°, namely the xR , left at 𝜃 = 90° is about 4.6% larger than that at 𝜃 = 30°. With regard to the inclination angle, it is almost constant for 𝜃 changing from 20° to 90°. Therefore, it can be concluded that, when the 𝜃 is increasing from 20° to 90°, the effect of varying 𝜃 on the right recirculation region is negligible. As shown in Fig. 20, for a given 𝜃 = 20°, 30°, and 50°, the left corner of the cavity is almost covered by the shear layer, which prevents produc-

Fig. 19. Inclination angle variation of the recirculation region(s) for different Da/D ratios.

The borderlines of boundary layer and shear layer for F1–F4 cases are extracted from Fig. 20 and shown in Fig. 21, where 𝛿 s and 𝛿 b of every case are marked. It is seen that, as the 𝜃 increases, the shear layer nearby the frontwall becomes thicker and moves upward gradually, whereas the shear layer in the right side of the cavity is affected slightly by the increasing 𝜃. The specific values of 𝛿 s and 𝛿 b for various 𝜃 are given in Fig. 22. It can find that both 𝛿 s and 𝛿 b increase gradually with the 𝜃,

Fig. 20. Mach number contours, borderlines of boundary layer and shear layer along with the recirculation regions for F1–F4 cases. 506

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Fig. 21. The boundary layer and shear layer for F1–F4 cases, the two isolines denote U /Ufr = 0.1 and 0.9, respectively.

Fig. 24. Streamwise length (–★–) and inclination angle (–●–) of the right recirculation region for the increasing 𝜃.

Fig. 22. Variation of 𝛿 s and 𝛿 b with increasing 𝜃.

ing the left recirculation region. Therefore, we speculate that the right recirculation region can also be eliminated by the rearwall inclination. For this purpose, the F1 and F2 cavities are modified by inclining their rearwalls with an angle of 20° and 30° respectively, and then the two new cavities are used to perform DSMC computations. The Mach number contour, streamlines along with the shear layer of every new cavity are presented in Fig. 25. By comparison with the F1 and F2 cases in Fig. 20, it is found that the right recirculation regions disappear completely due to the rearwall inclination and meanwhile the shear layer thickness is changed slightly. Therefore, a significant conclusion is that the recirculation region(s) within the cavity can be eliminated completely by the way of inclining the frontwall/rearwall with an appropriate angle.

efied hypersonic flow over a cavity with different geometric shapes, such as the length-to-depth ratio increasing from L/D = 1 to L/D = 8, the rearwall-to-frontwal height ratio (Da/D) of 0.5–2, and the inclined frontwall ranging from 𝜃 = 20° to 𝜃 = 70° Several evaluation parameters were proposed to quantitatively describe the flow characteristics for every case. Summarizing this study, some significant conclusions can be obtained, as follows: (1) For the rarefied hypersonic flow, the cavities can also be classified according to the flow characteristics of the shear layer. That is, for an open cavity, the shear layer spans the cavity, leads to a large recirculation region which occupies almost the whole area of the cavity. For a closed cavity, the shear layer impinges on the cavity floor, results in two smaller recirculation regions located at the left and right corners of the cavity, respectively. (2) The length-to-depth ratio (L/D) has an important influence on the flow characteristics of the shear layer and hence affects the recircu-

6. Conclusions In this paper, the direct simulation Monte Carlo (DSMC) method was employed to numerically investigate the flow characteristics of rar-

Fig. 23. Evolution of the (right) recirculation region with increasing 𝜃. 507

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Fig. 25. Mach number contour along with streamlines (top) and shear layer (bottom) for the cavity with inclined frontwall and rearwall. Note that there is no recirculation region within the cavity.

lation region. According to the computational results of B1–B8 cases, the cavity begins to transform into a closed type when the L/D = 6. For L/D > 6, the shear layer and recirculation region is changed slightly with the increase of L/D. A significant finding is that the criterion (i.e., L/D) for classifying the cavity type is not a constant but changes with freestream conditions. According to our currently computed results (see Fig. 10) and those in Ref [24], the criterion (i.e., L/D) would become smaller for the freestream with a higher altitude and a larger velocity. (3) For the cavity with a fixed L/D, its type can also be changed by adjusting the rearwall height. Specifically, the cavity would become an open type when the rearwall height increases, whereas it would become a closed type when the rearwall height decreases. The shear layer thickness decreases gradually with increasing Da/D ratio. (4) The left (right) recirculation region within the cavity can be eliminated completely by inclining the frontwall (rearwall) with an appropriate angle. If the frontwall and rearwall are simultaneously inclined with an appropriate angle (e.g., 20° or 30°), there would be no recirculation region within the cavity (see Fig. 25). That is, the frontwall /rearwall inclination is probably a simple but effective way to control the recirculation region(s) to disappear or reappear.

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