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Characterization of flow mixing and structural topology in supersonic planar mixing layer
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Characterization of flow mixing and structural topology in supersonic planar mixing layer Dongdong Zhang∗, Jianguo Tan∗∗, Liang Lv, Fei Li Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, Hunan, 410073, People's Republic of China
A R T I C LE I N FO
A B S T R A C T
Keywords: Aerospace propulsion system Supersonic planar mixing layer Mixing process Vortex topology Flow parameters Direct numerical simulation
The investigations on mixing process and structural topology properties of supersonic planar mixing layer with different inflow conditions are conducted by employing direct numerical simulation. First, the present highorder accuracy numerical methods are validated by comparing the simulation results with the data gained from previous well characterized experimental and numerical cases. Then the high-resolved three-dimensional numerical visualizations of supersonic mixing layer are presented by utilizing Q-criterion. The visualization results show the full development and evolution process of mixing layer, including the shear action, the transition process populated sequentially by Λ-vortices, hairpin vortices and braid structures and the establishment of selfsimilar turbulence. The effects of density ratio, velocity ratio and convective Mach number between the two parallel streams on mixing layer growth rate are evaluated by examining the indexes including velocity thickness and momentum thickness represented the mixing process. The results indicate that for the only variation of density ratio, the velocity thickness growth rates do not significantly vary, while the momentum thickness becomes larger when the upper and lower streams possess the same density. With the increase of only velocity ratio, the mixing layer becomes more stable and the velocity and momentum thickness are both drastically depressed in the whole flow field. As only convective Mach number increases, the mixing layer growth is inhibited in the near field through the transition delay of the flow, while in the far field, the growth rates are nearly the same for different convective Mach numbers. The spatial correlation analysis of structural topology indicates that the effects of each of the three main flow parameters on vortex topology lead to different mean structure sizes and shapes. The present research is useful for evaluating the effects of different flow parameters on mixing properties, which is important for the future scramjet combustor design and evaluation in engineering.
1. Introduction Supersonic planar mixing layer forming between two parallel streams with different velocities is one of the most important flow types which has been the focus of considerable research over the past three decades [1–3]. Due to its broad engineering applications, especially in supersonic combustion ramjet (scramjet) propulsion systems designed for hypersonic vehicles [4,5], extensive investigations have been carried out on physical mechanisms of planar mixing layers such as Kelvin–Helmholtz (K-H) instability and compressibility effects on the evolution of mixing layer growth rate. Since Brown and Roshko [6] first found large-scale coherent structures in planar mixing layers, researchers worldwide reached an agreement that large-scale coherent eddies are of crucial value in turbulent transport process and dominate the dynamical properties of mixing layers.
∗
Mixing process in supersonic mixing layer is largely dependent on compressibility effects [6]. It has been demonstrated that with the increase of compressibility, mixing layer growth rate is heavily inhibited and the turbulent intensities are drastically decreased [7,8]. To evaluate the compressibility effects on mixing layer, convective Mach number (Mc) was proposed by Papamoschou et al. [9] and Bogdanoff et al. [10]. Assuming equal specific heats, the convective Mach number can be expressed by Mc=(U1-U2)/(a1+a2), where U1 and U2 are the two inflow velocities, a1 and a2 are sound speeds for the two streams. Flow visualization results have shown that at low convective Mach number (Mc < 0.6), the flow structures possess similar behavior as that in incompressible flow, while at high convective Mach number (Mc > 0.6), due to three-dimensional primary instability with oblique instability waves, the mixing layer becomes highly three-dimensional [11].
Corresponding author. Corresponding author. E-mail addresses: [email protected] (D. Zhang), [email protected] (J. Tan).
∗∗
https://doi.org/10.1016/j.actaastro.2018.11.018 Received 25 September 2018; Received in revised form 6 November 2018; Accepted 13 November 2018 0094-5765/ © 2018 IAA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zhang, D., Acta Astronautica, https://doi.org/10.1016/j.actaastro.2018.11.018
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Nomenclature
Re Ru Rρ T U u' v' δ δnor ΔU
Abbreviations CFL CG DNS FG K–H scramjet TVD WENO
Courant–Friedrichs–Lewy coarse grid direct numerical simulation fine grid Kelvin–Helmholtz supersonic combustion ramjet total variation diminishing weighted essentially non-oscillatory
δω θ ρ Ψ
Variables <•> a M Mc P
root mean square sound speed, m/s Mach number, M = U/c convective Mach number, Mc=(U1−U2)/(a1+a2) pressure, kPa
Reynolds number velocity ratio, Ru=U2/U1 density ratio, Rρ = ρ2/ρ1 temperature, K velocity, m/s fluctuation of streamwise velocity, m/s fluctuation of transverse velocity, m/s velocity thickness, mm normalized spreading rate of mixing layers difference between the external velocities, ΔU = U1–U2, m/s vorticity thickness, mm momentum thickness, mm density, kg/m3 isoline value
Subscripts 1, 2
upper and lower flow sides
concerning the effects of Rρ and Ru on mixing process and structural evolution properties are carried out, the present investigations could provide explanations for the inconsistent conclusions drawn by former researchers during their experimental and numerical works.
During the past three decades, Mc has been an important non-dimensional parameter to evaluate mixing layer thickness and flow turbulent intensity. By utilizing Gaussian fitting, a relationship between non-dimensional mixing layer growth rate δnor and Mc 2 (δnor = 0.8e−3Mc + 0.2 ) was proposed by Dimotakis [12]. Following that, many experimental and numerical studies regarding mixing layer thickness and growth rate were conducted. Whereas, compared the experimental results of Clemens et al. [13] (Mc = 0.28) and Debisschop et al. [14] (Mc = 0.525), it can be found that the non-dimensional growth rate at Mc = 0.525 is much larger than that at Mc = 0.28, which is discrepant to the formula put forward by Dimotakis. It is obvious that the varying of only Mc leads to inconsistent conclusions. Besides, for turbulent intensity and Reynolds shear stress evaluating the flow fluctuation characteristics, two different conclusions have been drawn. Lau et al. [15] studied the effects of compressibility on turbulent statistics in weak compressibility conditions, and they concluded that although the growth rate of mixing layer decreases with the increase of Mc, the compressibility effects on velocity fluctuations and Reynolds stress tensor are not entirely determined. However, the velocity measurement results of Goebel et al. [16] showed that with the increase of Mc, a drastic decrease of both the velocity fluctuations and shear stress intensity occurred. Following that Gruber et al. [17] demonstrated the same conclusion. It is clearly that varying of only Mc failed to provide consistent conclusions regarding turbulence fluctuations in supersonic mixing layer flows. Evaluations of the flow parameters investigated in these previous studies indicate that the values of all three main parameters, density ratio Rρ = ρ2/ρ1, velocity ratio Ru=U2/U1, and Mc much varied from study to study. Thus, in the present research by employing direct numerical simulation method, supersonic planar mixing layers with different Rρ, Ru and Mc are analyzed respectively with focusing on mixing mechanisms and structural topology properties. After validation of the present numerical procedures, the visualization of full evolution process of the flow structures in supersonic planar mixing layer, including the formation of Λ-vortices, hairpin vortices and braid structures, breakdown of large-scale vortices and establishment of self-similar turbulence, is shown clearly in the simulation. Besides, two typical indexes, namely velocity thickness and momentum thickness are employed to evaluate the effects of the above three parameters variation on mixing process. To investigate the turbulence properties of the flow, the turbulent fluctuations under different flow conditions are analyzed. By utilizing spatial correlation analysis, the topology properties of vortex structures with different flow parameters are revealed. Since seldom research
2. Numerical method and validations 2.1. Governing equations and numerical scheme To obtain the computational results, the unsteady, three-dimensional compressible Navier–Stokes equations are used as governing equations for supersonic spatially developing mixing layer. For the simulation of compressible flow, the use of low-dissipation, high-order shock-capturing schemes is an essential ingredient. The objective is to avoid excessive numerical damping of the flow features over a wide range of length scales as well as to prevent spurious numerical oscillations near shock waves and discontinuities, and the family of Weighted Essentially Non-Oscillatory (WENO) schemes can be a good choice to achieve this goal. In the present research, the governing equations are solved by employing our in-house newly developed simulation codes this year. For approximation of inviscid fluxes, the fifth-order WENO scheme has been used. For the WENO scheme formulation according to Jiang and Shu [18,19], formally odd order of N = 5 and an 5-point stencil is employed to calculate the inviscid numerical fluxes. Since that the fifthorder WENO scheme has been successfully applied to numerous simulations of various strong compressible flows, we do not present the details in this paper. The central difference scheme of sixth-order is used for the discretization of viscous fluxes. The discretized equations are integrated in time by means of explicit third-order total variation diminishing (TVD) Runge-Kutta algorithm. The time step is determined according to a proper Courant–Friedrichs–Lewy (CFL) number, which is set as 0.5 in this paper.
2.2. Boundary conditions In the case of a mixing layer, the velocity profiles, when scaled in the crosswise direction with a length proportional to the mixing layer width, collapse into a single profile, which is close to a tanh function [20]. Therefore, for the inflow boundary conditions, the mean streamwise inlet velocity distribution is specified as a tanh-type profile, which is the most common model for analyzing mixing layers [21]: 2
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U(0) =
U1 + U2 2
−
U1 − U2 2
tanh
exhibit a Gaussian shape centered on the mixing layer centerline (Y* = 0) where flow turbulent activity is the most intense. The halfwidth of the transverse profiles are in particular accurately reproduced by the simulations. Apart from that, a test case is calculated with the same parameters except for the coarser computational mesh to examine the level of grid dependency. The cell size is about one and a half times larger than that for the fine grid case. It can be seen that compared to the case with coarse grids (CG), the calculation with fine grids (FG) show better agreements with the experimental data. To further confirm the grid independence of the present numerical procedures, comparison of momentum thickness versus X direction in coarse and fine grid levels is presented in Fig. 3. It can be seen that though a slight difference, the results obtained in the two grid sets agree well with each other. Therefore in this paper, the computational mesh with fine grid sets possesses property of high-resolution and can be employed in the present numerical study.
( ) y 2θ (0)
V(0) = 0 W(0) = 0 for three − dim ensional cases
(1)
Where θ (0) is the initial mixing layer momentum thickness at the inlet position. For the expression of U(0), the first term indicates the mean flow. The last term describes a smoothly varying flow within a transition layer with width 2θ (0) . To excite the flow instability and seed the laminar breakdown of the mixing layer, the flow is excited at the upstream boundary of the calculation domain. The disturbance function of transverse-velocity fluctuations is set as:
f1′ = AΔUG (y) sin(2πfv t + φ)
(2)
Here, A is the amplitude, G(y) is Gaussian function, φ is the random phase ranging from 0 to 2π, fv is the most unstable frequency of the mixing layer, ΔU is velocity difference and ΔU = U1 − U2 , t is the time. Apart from that, for three-dimensional cases, to obtain natural happened mixing layers, the supersonic mixing layer is periodically excited by a superimposed three-dimensional oblique waves with the following form,
2.4. Precision estimates For large scale simulations of complex compressible flows, estimating precision and errors accumulation is necessary [26]. Errors accumulation takes place for successive time steps, and it should be evaluated for each simulation case. Here we employed the method used by Smirnov et al. [27] to evaluate the error accumulation and characterize the reliability of the simulation results. Detailed description of the method can be found in Ref. [27]. Table 2 presents the accumulation of error for different grid resolutions and physical time simulations using the present numerical procedures. 100000 time steps are calculated for both cases to ensure the accuracy of the turbulent statistical results. It can be seen that for coarse grid case, the accumulation of errors takes place in a slightly faster manner, while decreases with the increase of grid resolution. Besides, the results demonstrate that the present fine grid set and numerical procedures can provide high reliability for the study.
f2′ = B(y) (cos(± βz + ωt ) + cos( ± βz + 0.5ωt ) + cos( ± βz + 0.25ωt )) (3) Here, B(y) is Gaussian Function, ω corresponds to the most unstable frequency obtained from the linear theory proposed by Michalke [21], and ω = 2πfvt. Sandham et al. [22] have demonstrated that simulations with forcing of linear instability waves can produce the development of large-scale structures similar to a fully nonlinear computation with a random initial condition. β is the wave numbers in spanwise direction, and its value can be obtained in Sandham et al.[ 22]. The supersonic outflow boundary condition is imposed at the transverse boundaries in Y direction, since the reflection waves are not expected to reflect back to affect the calculating main flows. At the outlet of the calculating domain, the boundary condition is also set as outflow boundary condition, which has been discussed and demonstrated in former research [23]. Besides, for three-dimensional cases, the periodical boundary condition is enforced in spanwise (Z) directions. The detailed three-dimensional computational model is presented in Fig. 1.
3. Results and discussions 3.1. Visualization and analysis of flow structures Since it has been demonstrated by Martha et al. [28] that using isosurface of vortex intense is a good way to visualize the structures in the flow, we first employed Q-criterion to detect the vortex structures and analyze the mixing characteristics of supersonic mixing layer. Fig. 4 depicts the three-dimensional visualization of supersonic spatially developing mixing layer with Mc = 1.0. The variable Q is defined as the second invariant of the velocity gradient tensor and it is well suited to vortex-field identification [29]. The iso-surface of the vortex structures with a non-dimensional Q value of 0.0005 is presented in Fig. 4. The color changes from red to blue as Y increases. It can be clearly seen that large-scale structures and the evolution process are well captured through the present numerical scheme. After a period of shear action of upper and lower streams near the inlet, typical K-H
2.3. Grid specification and numerical validation The grids in the mixing layer area are highly refined and the corresponding shear layer vortex structures are well resolved. In Y direction, the grids are clustered towards the centerline to provide a proper resolution of the mixing layer. For two-dimensional cases, the grid points in X direction are uniform spaced, and for three-dimensional cases, in Z directions the grid points are uniform spaced as well. The computational domain size is defined as Lx × Ly × Lz = 240 × 60 × 15 mm3 with the fine grid sets of Nx × Ny × Nz = 1440 × 240 × 90. The minimum grid space ΔXmin, ΔYmin, ΔZmin is 0.167 mm, 0.08 mm and 0.167 mm, respectively. The Reynolds number based on the initial vorticity thickness is about Re(δω) = 400. To validate the present numerical schemes, the comparison between the numerical results and former experimental and numerical data are conducted. We carried out the numerical simulation of a classical compressible mixing layer model, which has been experimentally investigated by Goebel and Dutton [16]. The flow parameters are listed in Table 1. Fig. 2 presents the comparison of distribution of mean velocity and streamwise turbulent fluctuations compared with the experimental data and some recent DNS results [24,25]. It can be seen that in self-similar region of the flow, the agreement between numerical and experimental data is good. Moreover, for velocity fluctuations, the simulation profiles
Fig. 1. Schematic of the computational model. 3
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vortex structures namely Λ structures roll up in transition region due to the initial K-H instability. Since it has been demonstrated by Sandham et al. [11] that three-dimensional instability plays a dominate role in the supersonic mixing flow when Mc is larger than 0.6, in the present simulation (Mc = 1.0) after a short distance in transition region, Λ structures have a tendency to develop in transverse direction to form three dimensional structures. Apart from that, another 3-D test case using a coarser grid set is conducted to verify the grid independence of the flow structures prior to transition to turbulence, as shown in Fig. 4(b) it can be found that with coarser grid set the evolution of three-dimensional structures are clearly revealed in the instantaneous visualization result as well, showing that the present fine grid set is suitable for the next step study. To clearly illustrate the evolution properties, Fig. 5 depicts more details of the structures with Q = 0.001 in transition region. Due to three-dimensional instability, the Λ vortices evolve into a new structure type called hairpin structures quickly. The flow regions in the dotted boxes A, B, C and D clearly show the evolution of hairpin vortices in the present mixing layer. The two legs of hairpin structure tend to rotate and distort, and eventually breakdown into small scale slender structures. The head of the hairpin vortices experiences instability and stretches and rises in transverse direction. Since the stretch and breakdown process can improve the transport of momentum and scalar of upper and lower streams [30,31], the development and evolution of hairpin vortex structures play a dominant role in the mixing process of the flow. Besides, it should be noted that the preponderance of hairpinlike structures has recently been reported in DNS of a spatially developing boundary layer by Ringuette et al. [32]. Both their research and our findings indicate that hairpin-structures play an important role in the breakdown of shear flows.
Table 1 The flow parameters of experiments conducted by Goebel and Dutton [16].
Upper stream Lower stream
U/(m/s)
M
ρ/(kg/m3)
P/(kPa)
Mc
519 409
2.04 1.4
1.0 0.76
46 46
0.2
3.2. Effects of density ratios variation To highlight the mixing mechanisms affected by the variation of Rρ, the velocity ratio Ru and convective Mach numbers Mc are set as constant, and the inflow parameters are listed in Table 3. To evaluate the mixing properties of supersonic mixing layers, turbulent statistical analysis is necessary. Fig. 6 depicts the normalized mean streamwise velocity isolines of different density ratios (Rρ = 0.75, 1.0, and 1.5 respectively). The isoline value Ψ is calculated based on the following formulation:
Ψ = (U − U2)/ ΔU
Fig. 2. Distribution of mean velocity and streamwise turbulent intensity.
(4)
With the development of the mixing layer, the distance between the transverse locations where Ψ = 0.1 and Ψ = 0.9 show character of linear increase. Meanwhile, the evolution of mixing layer has a tendency of inclination to the side of which the velocity is lower, this phenomenon is due to that the engulfment of upper and lower streams is asymmetric when the inflow velocity is different, and the reason will be discussed in detail in the following section. To quantitatively analyze the mixing layer growth rate of the flow, the velocity thickness is defined based on the velocity isolines,
δ = YU0.9 − YU0.1
(5)
Here, YU0.9 and YU0.1 denote the transverse coordinates where U = U1 − 0.1ΔU and U = U2 + 0.1ΔU , respectively. The velocity thickness versus X is presented in Fig. 7. It is clearly observed that with the flow downstream the streamwise direction, the influence of the change of density ratio on the velocity thickness growth rate is slight. As Rρ increases, the degree of that thickening is not significant as a whole. Here we concluded that Rρ has no obvious effect on the evolution of velocity thickness. This conclusion is consistent with the observations obtained in the experiments of Brown and Roshko [6]. Whereas, the momentum thickness under various Rρ conditions shows different evolution properties, as presented in Fig. 8. It can be seen that although a linear growth tendency is observed for all the three
Fig. 3. Comparison of momentum thickness evolution with different grid resolutions.
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Table 2 Error estimates. Allowable error 5% 5%
Grid resolution 960 × 240 × 90 1440 × 240 × 90
Physical time simulated (ms) 2.5 2.5
Number of time steps 100000 100000
Accumulated error 0.0000000005950405 0.0000000005950401
Allowable number of time steps 10
Reliability 2.6571952·105 2.6571969·105
2.6571952·10 2.6571969·1010
Table 3 The flow parameters with the varying of Rρ.
Ru Mc Rρ
Case 1
Case 2
Case 3
0.6 0.65 0.75
0.6 0.65 1.0
0.6 0.65 1.5
Fig. 4. Instantaneous overall visualization of vortex structures with Mc = 1.0. (a) Fine grid case, (b) Coarse grid case. Fig. 6. The distribution of normalized streamwise mean velocity isolines of different Rρ: (a) Ru = 0.6, Mc = 0.65, Rρ = 0.75, (b) Ru = 0.6, Mc = 0.65, Rρ = 1.0, (c) Ru = 0.6, Mc = 0.65, Rρ = 1.5.
cases, at the condition of Rρ = 1.0, the growth rate of the momentum thickness is slightly higher than that of the two streams with different densities. It is obvious that the influence of density ratio cannot be ignored. It has been demonstrated by Dimokatis et al. [33] that for compressible mixing layers with the same inflow densities, the engulfment and entrainment of the flow possess characteristic of
symmetry, and the symmetry can help to increase the level of the flow intensity, which contributes much to the higher growth rate of momentum thickness in fully turbulent region.
Fig. 5. Detailed view of vortex structures in transition region of Mc = 1.0 with iso-surface of Q = 0.001. 5
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rises with more fluctuations. Whereas, at Ru = 0.6 and 0.75 the flow is more stable, the fluctuations are slight and the thickness increase linearly. 3.4. Effects of convective Mach numbers variation Although a consensus has been reached that the increase of Mc would increase the flow compressibility while decrease the mixing layer growth rate, here to make comparisons, it is still essential to make a detailed and quantitative analysis of the influence of Mc on mixing properties. Table 5 lists the inflow parameters with Rρ and Ru fixed as constant. Fig. 12 presents the normalized streamwise mean velocity isolines (Ψ = 0.1, 0.5, and 0.9) of different Mc. Due to the compressibility effects, with the increase of Mc, the mixing layer experiences longer shear action distance. At the condition of Mc = 1.0, the distance for the shear action is about 80 mm, and the transition of the flow is heavily delayed. Here we concluded that with the increase of Mc, the effect on the suppression of rolling up process of K-H structures contributes much to the decrease of mixing layer growth rate. This conclusion can be quantitatively drawn from the analysis of velocity and momentum thickness evolutions, as shown in Figs. 13 and 14. In the near field where the shear action occurs, the values of δ and θ both decrease monotonically with the increase of Mc, indicating that high compressibility inhibits the mixing layer growth rate through the transition delay of the flow. Apart from that, it should be noted that in the far flow field at X > 140 mm, the fitting linear growth rate for all the three cases are nearly the same when keeping Rρ and Ru constant. Going back to the inconsistent conclusion drawn by Clemens et al. [13] and Debisschop et al. [14], it can be easily understood from the present research that for different Rρ and Ru conditions, the variation of only Mc may lead to different conclusions.
Fig. 7. The distribution of velocity thickness of different Rρ.
3.5. Turbulent intensity analysis The Reynolds stresses in the mixing layer are found to be closely associated with mixing process and the vortex structures. For free shear layers, the flow will enter into a self-similar status in fully developed region, meaning that both mean variables and turbulence Reynolds stresses behave in a self-similar manner [34,35]. Thus in the present section, to reveal the effects of different flow parameters on the mixing process, the distribution of Reynolds stresses < u′v′ > /ΔU 2 at streamwise position of X = 0.95Lx under different Rρ, Ru and Mc are researched. It should be noted to show the property of profiles collapse, the transverse coordinate is not normalized by local mixing layer thickness in the present investigations. Fig. 15 depicts the distribution of Reynolds stresses with different Rρ. Though possessing similar shape, the peak values of Reynolds stresses are apparently larger in the core mixing region when the upper and lower streams have the same density, resulting in more intense of turbulent fluctuations. Larger turbulent fluctuation intensity will drastically promote the flow instability and strengthen the flow three-dimensionality, and more large-scale structures will breakdown into small-scale vortices, which can increase the interface area and increase the mixing level. Therefore, as discussed above in Fig. 8, in self-similar region the mixing layer growth rate at Rρ = 1.0 is slightly larger, which shows well agreement with the Reynolds stresses distributions here.
Fig. 8. The distribution of momentum thickness of different Rρ.
3.3. Effects of velocity ratios variation In this section the variables Rρ and Mc are held to be constant to analyze the effects of Ru on the mixing process. Three cases with different velocity ratios are calculated and Table 4 lists the calculation parameters. Fig. 9 depicts the normalized streamwise mean velocity isolines (Ψ = 0.1, 0.5, and 0.9) of different Ru. With the increase of Ru, the distance between Ψ = 0.1 and 0.9 decrease drastically, indicating that Ru has great influence on the mixing layer growth rate. To better analyze the effects, Fig. 10 presents the velocity thickness versus X of three Ru conditions. It can be revealed clearly that the increase of Ru leads to an obvious reduction of the velocity thickness. Fig. 11 illustrates the observed variation of momentum thickness along the streamwise direction of the flow. At Ru = 0.45 and 0.6, the flow becomes instability quickly, while for the flow at Ru = 0.75, the flow experiences a long distance of shear action (40 mm approximately) and then K-H vortices roll up and the momentum thickness rises at a much lower rate. The results revealed that although with the same Mc, the variation of Ru has important influence on the mixing process. Meanwhile, it can be seen that due to the big velocity difference, the flow possesses much intensity at Ru = 0.45, and the momentum thickness
Table 4 The flow parameters with the varying of Ru.
Ru Mc Rρ
6
Case 1
Case 2
Case 3
0.45 0.65 1.0
0.6 0.65 1.0
0.75 0.65 1.0
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Fig. 11. The distribution of momentum thickness of different Ru. Table 5 The flow parameters with the varying of Mc.
Fig. 9. The distribution of normalized streamwise mean velocity isolines of different Ru: (a) Mc = 0.65, Rρ = 1.0, Ru = 0.45, (b) Mc = 0.65, Rρ = 1.0, Ru = 0.6, (c) Mc = 0.65, Rρ = 1.0, Ru = 0.75.
Rρ Ru Mc
Case 1
Case 2
Case 3
0.75 0. 6 0.4
0.75 0.6 0.65
0.75 0.6 1.0
Fig. 10. The distribution of velocity thickness of different Ru.
Fig. 16 shows the distribution of Reynolds stresses in transverse direction at different Ru conditions. It can be found that in the external part of the mixing layer at Ru = 0.75, the value of Reynolds stress is apparently lower, indicating that the mixing zone where turbulent fluctuation occurs is much smaller than the cases of Ru = 0.46 and Ru = 0.6. The same phenomenon is also appeared in Fig. 17. It can be seen that when the flow is at highly compressible conditions of Mc = 1.0, the fluctuation is heavily inhibited, leading to observed discrepancies between these three cases. Apart from that, as mentioned in section 3.2, it is interesting to point out that the mixing region has the tendency to incline to the incoming stream side with low velocity. Especially with smaller velocity ratios at Ru = 0.45 and Ru = 0.6, the phenomenon is more notable. The results, though first shown in the present paper, have been reported by Konrad et al. [36] in their investigations on incompressible shear layers. The reason is that the entrainment and engulfment of upper and lower streams are not symmetrical due to the velocity difference of the
Fig. 12. The distribution of normalized streamwise mean velocity isolines of different Mc: (a) Rρ = 0.75, Ru = 0.6, Mc = 0.4, (b) Rρ = 0.75, Ru = 0.6, Mc = 0.65, (c) Rρ = 0.75, Ru = 0.6, Mc = 1.0.
streams. To quantitatively analyze the influence of different flow parameters on turbulent statistical properties, Table 6 presents the peak intensities. It can be noted that though slight variation, the effects of Rρ and Ru on the peak intensities are not significant, while with the increase of Mc, the peak values of Reynolds stresses experience sharp decrease.
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Fig. 16. Reynolds stresses distribution of different Ru.
Fig. 13. The distribution of velocity thickness of different Mc.
Fig. 14. The distribution of momentum thickness of different Mc.
Fig. 17. Reynolds stresses distribution of different Mc.
results, the correlation coefficient Cuu in X-Y plane can be defined by the following formula,
Cuu (X0 + ΔX , Y0 + ΔY ) =
u′ (X0 , Y0)⋅u′ (X0 + ΔX , Y0 + ΔY ) u′ (X0 , Y0)2 ⋅ u′ (X0 + ΔX , Y0 + ΔY )2
(6)
Where (X0, Y0) is reference location, u' is the streamwise fluctuation velocity. Poggie et al. [37] have demonstrated that a set of more than 200 instantaneous images can ensure the stability of the calculating results. To analyze the vortex topology characteristics in self-similar turbulent region, the reference location is chosen at (192 mm, 30 mm) far downstream the flow field. Fig. 18 depicts the spatial correlation of flow structures versus Rρ variation. The correlation contours are seen to be approximately elliptical in shape with the major axis aligned with the flow direction, which shows well agreement with former results [37]. Bourdon et al. [38] have demonstrated that the shape of correlation contours represents the mean size of the flow structures. Thus, to quantitatively analyze the vortex topology properties, the least square ellipse fitting method is used to gain the size of the vortices. The region enclosed by contour level of 0.5 has principal axes of d1 = 7.348 mm, 7.932 mm, 7.857 mm and d2 = 3.446 mm, 3.398 mm, 3.721 mm for the cases of Rρ = 0.75, 1.0 and 1.5, respectively. Though possessing a slight larger size for the condition of Rρ = 1.0, the effects of Rρ variation on the size of flow structures are not significant. Whereas, at Rρ = 1.0 the
Fig. 15. Reynolds stresses distribution of different Rρ.
3.6. Vortex topology analysis To investigate the vortex topology characteristics, the spatial correlation analysis method is employed. Based on the present simulation 8
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Table 6 Comparison of peak Reynolds stresses with different inflow parameters. Ru=0.6, Mc=0.65 〈u'v'〉/△U
2
Rρ = 0.75 0.05595
Mc = 0.65, Rρ = 1.0 Rρ = 1.0 0.06539
Rρ = 1.5 0.05658
Ru = 0.45 0.0602
Ru = 0.6 0.06539
Rρ = 0.75, Ru = 0.6 Ru = 0.75 0.05461
Mc = 0.4 0.07277
Mc = 0.65 0.05595
Mc = 1.0 0.04391
Fig. 18. Spatial correlation of flow structures versus Rρ: (a) Ru = 0.6, Mc = 0.65, Rρ = 0.75, (b) Ru = 0.6, Mc = 0.65, Rρ = 1.0, (c) Ru = 0.6, Mc = 0.65, Rρ = 1.5.
Fig. 19. Spatial correlation of flow structures versus Ru: (a) Mc = 0.65, Rρ = 1.0, Ru = 0.45, (b) Mc = 0.65, Rρ = 1.0, Ru = 0.6, (c) Mc = 0.65, Rρ = 1.0, Ru = 0.75.
Fig. 20. Spatial correlation of flow structures versus Mc: (a) Rρ = 1.0, Ru = 0.6, Mc = 0.4, (b) Rρ = 1.0, Ru = 0.6, Mc = 0.65, (c) Rρ = 1.0, Ru = 0.6, Mc = 1.0.
short axis reach the minimum of 5.146 mm and 2.322 mm. Going back to Figs. 10 and 11, it can be seen the structure topology evolution corresponds well to the mixing process of different flow parameters. The smaller sizes of the mean vortex structures are responsible for the inhibition of the mixing growth rate at higher Ru condition. The spatial correlations of flow structures versus Mc are presented in Fig. 20. An interesting phenomenon is that as Mc increases, the
shape of the mean structure has the tendency to be compressed and become narrow, since the ratio of long axis and short axis is 2.334, which is much larger than that of the other two cases. Fig. 19 shows the correlation plots for the cases with different Ru. It can be obviously seen that with the increase of velocity ratios, though having a shape similar to an ellipse, the size of mean flow structures experiences sharp decrease. When Ru is fixed to 0.75, the long axis and 9
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influences on the size of flow structures are not significant, while the shape of the mean structure tend to be compressed and become narrow at Rρ = 1.0 condition. For Ru variation, the size of mean flow structures experiences sharp decrease with the increase of Ru, and the smaller size of the mean vortex structures are responsible for the inhibition of the mixing growth rate at higher Ru condition. As Mc increases, the vortex size experiences decrease but not significant. The present investigations could provide explanations for the inconsistent conclusions drawn by former researchers during their experimental and numerical works.
vortex size experiences decrease but not significant. Going back to Figs. 13 and 14, it can be found that the phenomenon is reasonable since that in far flow field region the fitting linear growth rates for the three different cases are nearly the same when keeping Rρ and Ru constant. Therefore, although Mc has been demonstrated to be an important non-dimensional parameter to evaluate the flow compressibility, the varying of only Mc usually leads to inconsistent conclusions in previous investigations. The effects of each of the three main flow parameters, including Rρ, Ru, and Mc, on mixing process and vortex topology evolution are different, and it is necessary to analyze the influence of each parameter respectively.
Acknowledgments 4. Conclusions and remarks The authors would like to express their thanks for the support from National Natural Science Foundation of China (Grant Nos. 11272351, and 91441121), and the Graduate Student Research Innovation Project of Hunan Province (Grant No. CX2016B001).
In the present paper, direct numerical simulations of spatially developing supersonic planar mixing layers are conducted with focusing on the effects of varying of three main flow parameters including Rρ, Ru and Mc on the mixing process and vortex topology properties of the flow. After validation of the numerical procedures, two indexes, namely velocity thickness and momentum thickness are employed to evaluate the mixing characteristics with different inflow parameters. The distribution of Reynolds stresses are analyzed to show the turbulent intensities. The spatial correlation analysis is carried out to investigate the vortex topology properties influenced by different flow parameters. The following conclusions can be drawn based on the results:
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(1) The comparison between present numerical results and former experimental and numerical data reaches well agreements, indicating that the present numerical method could efficiently capture the characteristics of compressible mixing layer flow. The profiles show a Gaussian shape centered on the mixing layer centerline where flow turbulent activity is the most intense. The half-width of the transverse profiles are in particular accurately reproduced by the simulations. (2) The overall development and evolution of three-dimensional vortex structures in supersonic planar mixing layers are well reproduced. The mixing layer experiences shear action region, turbulent transition region and fully turbulent region downstream the flow field. In transition region, typical flow structures, including Λ vortices, hairpin structures and braid vortices are well captured, and the role of hairpin vortices played in mixing process of shear layers is analyzed in detail. (3) The effect of Rρ variation on the velocity thickness growth rate is not significant, while the momentum thickness growth rate becomes slightly higher when the inflows have the same density. With the increase of Ru, the indexes evaluating mixing efficiency including velocity and momentum thickness are both heavily depressed. However, for the variation of Mc, the mixing layer experiences longer shear action distance at Mc = 1.0 and the transition of the flow is heavily delayed, which contributes much to the low growth rate in the near flow field. Besides, in the far flow field at X > 140 mm, the fitting linear growth rate for all the three cases are nearly the same when keeping Rρ and Ru constant. (4) The effects of each of the three main flow parameters on the Reynolds stresses are different. The peak values of Reynolds stresses in self-similar region experience sharp decrease with the increase of Mc. Whereas, for the cases of varying of Rρ and Ru, the change of peak values of Reynolds stresses is slight. The smaller Reynolds stresses in transverse region, the less intense of turbulent fluctuations, which is responsible for the inhibition of mixing process. The mixing region has the tendency to incline to the incoming stream side with low velocity. The results are first shown in the present paper and the reasons are analyzed. (5) The spatial correlation analysis results show that the effects of each of the three main flow parameters on vortex topology lead to different mean structure sizes and shapes. For Rρ variation, the 10
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discontinuous Galerkin method, Acta Mech. Sin. 27 (2010) 318–329. [35] D. Oster, I. Wygnanski, The forced mixing layer between parallel streams, J. Fluid Mech. 113 (1982) 91–130. [36] J.H. Konrad, An Experimental Investigation of Mixing in Two-dimensional Turbulent Shear Flows with Applications to Diffusion - Limited Chemical Reactions, Ph.D. Thesis California Institute of technology, 1976. [37] J. Poggie, P.J. Erbland, A.J. Smits, R.B. Miles, Quantitative visualization of compressible turbulent shear flows using condensate-enhanced Rayleigh scattering, Exp. Fluid 37 (2004) 438–454. [38] C.J. Bourdon, J.C. Dutton, Planar visualizations of large-scale turbulent structures in axisymmetric supersonic separated flows, Phys. Fluids 11 (1999) 201–213.
[29] Y. Dubief, F. Delcayre, On coherent-vortex identification in turbulence, J. Turbul. 1 (2000) 11–23. [30] D.D. Zhang, J.G. Tan, J.W. Hou, Structural and mixing characteristics influenced by streamwise vortices in supersonic flow, Appl. Phys. Lett. 110 (2017) 124101. [31] D.D. Zhang, J.G. Tan, H. Li, J.W. Hou, Fine flow structure and mixing characteristic in supersonic flow induced by a lobed mixer, Acta Phys. Sin. 66 (2017) 104702. [32] M.J. Ringuette, M. Wu, M.P. Martin, Coherent structures in direct numerical simulation of turbulent boundary layers at Mach 3, J. Fluid Mech. 594 (2008) 59–69. [33] P.E. Dimotakis, Two-dimensional shear layer entrainment, AIAA J. 24 (1986) 1791–1796. [34] X.T. Shi, J. Chen, Numerical simulations of compressible mixing layers with a
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Characterization of flow mixing and structural topology in supersonic planar mixing layer Dongdong Zhang∗, Jianguo Tan∗∗, Liang Lv, Fei Li Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, Hunan, 410073, People's Republic of China
A R T I C LE I N FO
A B S T R A C T
Keywords: Aerospace propulsion system Supersonic planar mixing layer Mixing process Vortex topology Flow parameters Direct numerical simulation
The investigations on mixing process and structural topology properties of supersonic planar mixing layer with different inflow conditions are conducted by employing direct numerical simulation. First, the present highorder accuracy numerical methods are validated by comparing the simulation results with the data gained from previous well characterized experimental and numerical cases. Then the high-resolved three-dimensional numerical visualizations of supersonic mixing layer are presented by utilizing Q-criterion. The visualization results show the full development and evolution process of mixing layer, including the shear action, the transition process populated sequentially by Λ-vortices, hairpin vortices and braid structures and the establishment of selfsimilar turbulence. The effects of density ratio, velocity ratio and convective Mach number between the two parallel streams on mixing layer growth rate are evaluated by examining the indexes including velocity thickness and momentum thickness represented the mixing process. The results indicate that for the only variation of density ratio, the velocity thickness growth rates do not significantly vary, while the momentum thickness becomes larger when the upper and lower streams possess the same density. With the increase of only velocity ratio, the mixing layer becomes more stable and the velocity and momentum thickness are both drastically depressed in the whole flow field. As only convective Mach number increases, the mixing layer growth is inhibited in the near field through the transition delay of the flow, while in the far field, the growth rates are nearly the same for different convective Mach numbers. The spatial correlation analysis of structural topology indicates that the effects of each of the three main flow parameters on vortex topology lead to different mean structure sizes and shapes. The present research is useful for evaluating the effects of different flow parameters on mixing properties, which is important for the future scramjet combustor design and evaluation in engineering.
1. Introduction Supersonic planar mixing layer forming between two parallel streams with different velocities is one of the most important flow types which has been the focus of considerable research over the past three decades [1–3]. Due to its broad engineering applications, especially in supersonic combustion ramjet (scramjet) propulsion systems designed for hypersonic vehicles [4,5], extensive investigations have been carried out on physical mechanisms of planar mixing layers such as Kelvin–Helmholtz (K-H) instability and compressibility effects on the evolution of mixing layer growth rate. Since Brown and Roshko [6] first found large-scale coherent structures in planar mixing layers, researchers worldwide reached an agreement that large-scale coherent eddies are of crucial value in turbulent transport process and dominate the dynamical properties of mixing layers.
∗
Mixing process in supersonic mixing layer is largely dependent on compressibility effects [6]. It has been demonstrated that with the increase of compressibility, mixing layer growth rate is heavily inhibited and the turbulent intensities are drastically decreased [7,8]. To evaluate the compressibility effects on mixing layer, convective Mach number (Mc) was proposed by Papamoschou et al. [9] and Bogdanoff et al. [10]. Assuming equal specific heats, the convective Mach number can be expressed by Mc=(U1-U2)/(a1+a2), where U1 and U2 are the two inflow velocities, a1 and a2 are sound speeds for the two streams. Flow visualization results have shown that at low convective Mach number (Mc < 0.6), the flow structures possess similar behavior as that in incompressible flow, while at high convective Mach number (Mc > 0.6), due to three-dimensional primary instability with oblique instability waves, the mixing layer becomes highly three-dimensional [11].
Corresponding author. Corresponding author. E-mail addresses: [email protected] (D. Zhang), [email protected] (J. Tan).
∗∗
https://doi.org/10.1016/j.actaastro.2018.11.018 Received 25 September 2018; Received in revised form 6 November 2018; Accepted 13 November 2018 0094-5765/ © 2018 IAA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zhang, D., Acta Astronautica, https://doi.org/10.1016/j.actaastro.2018.11.018
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Nomenclature
Re Ru Rρ T U u' v' δ δnor ΔU
Abbreviations CFL CG DNS FG K–H scramjet TVD WENO
Courant–Friedrichs–Lewy coarse grid direct numerical simulation fine grid Kelvin–Helmholtz supersonic combustion ramjet total variation diminishing weighted essentially non-oscillatory
δω θ ρ Ψ
Variables <•> a M Mc P
root mean square sound speed, m/s Mach number, M = U/c convective Mach number, Mc=(U1−U2)/(a1+a2) pressure, kPa
Reynolds number velocity ratio, Ru=U2/U1 density ratio, Rρ = ρ2/ρ1 temperature, K velocity, m/s fluctuation of streamwise velocity, m/s fluctuation of transverse velocity, m/s velocity thickness, mm normalized spreading rate of mixing layers difference between the external velocities, ΔU = U1–U2, m/s vorticity thickness, mm momentum thickness, mm density, kg/m3 isoline value
Subscripts 1, 2
upper and lower flow sides
concerning the effects of Rρ and Ru on mixing process and structural evolution properties are carried out, the present investigations could provide explanations for the inconsistent conclusions drawn by former researchers during their experimental and numerical works.
During the past three decades, Mc has been an important non-dimensional parameter to evaluate mixing layer thickness and flow turbulent intensity. By utilizing Gaussian fitting, a relationship between non-dimensional mixing layer growth rate δnor and Mc 2 (δnor = 0.8e−3Mc + 0.2 ) was proposed by Dimotakis [12]. Following that, many experimental and numerical studies regarding mixing layer thickness and growth rate were conducted. Whereas, compared the experimental results of Clemens et al. [13] (Mc = 0.28) and Debisschop et al. [14] (Mc = 0.525), it can be found that the non-dimensional growth rate at Mc = 0.525 is much larger than that at Mc = 0.28, which is discrepant to the formula put forward by Dimotakis. It is obvious that the varying of only Mc leads to inconsistent conclusions. Besides, for turbulent intensity and Reynolds shear stress evaluating the flow fluctuation characteristics, two different conclusions have been drawn. Lau et al. [15] studied the effects of compressibility on turbulent statistics in weak compressibility conditions, and they concluded that although the growth rate of mixing layer decreases with the increase of Mc, the compressibility effects on velocity fluctuations and Reynolds stress tensor are not entirely determined. However, the velocity measurement results of Goebel et al. [16] showed that with the increase of Mc, a drastic decrease of both the velocity fluctuations and shear stress intensity occurred. Following that Gruber et al. [17] demonstrated the same conclusion. It is clearly that varying of only Mc failed to provide consistent conclusions regarding turbulence fluctuations in supersonic mixing layer flows. Evaluations of the flow parameters investigated in these previous studies indicate that the values of all three main parameters, density ratio Rρ = ρ2/ρ1, velocity ratio Ru=U2/U1, and Mc much varied from study to study. Thus, in the present research by employing direct numerical simulation method, supersonic planar mixing layers with different Rρ, Ru and Mc are analyzed respectively with focusing on mixing mechanisms and structural topology properties. After validation of the present numerical procedures, the visualization of full evolution process of the flow structures in supersonic planar mixing layer, including the formation of Λ-vortices, hairpin vortices and braid structures, breakdown of large-scale vortices and establishment of self-similar turbulence, is shown clearly in the simulation. Besides, two typical indexes, namely velocity thickness and momentum thickness are employed to evaluate the effects of the above three parameters variation on mixing process. To investigate the turbulence properties of the flow, the turbulent fluctuations under different flow conditions are analyzed. By utilizing spatial correlation analysis, the topology properties of vortex structures with different flow parameters are revealed. Since seldom research
2. Numerical method and validations 2.1. Governing equations and numerical scheme To obtain the computational results, the unsteady, three-dimensional compressible Navier–Stokes equations are used as governing equations for supersonic spatially developing mixing layer. For the simulation of compressible flow, the use of low-dissipation, high-order shock-capturing schemes is an essential ingredient. The objective is to avoid excessive numerical damping of the flow features over a wide range of length scales as well as to prevent spurious numerical oscillations near shock waves and discontinuities, and the family of Weighted Essentially Non-Oscillatory (WENO) schemes can be a good choice to achieve this goal. In the present research, the governing equations are solved by employing our in-house newly developed simulation codes this year. For approximation of inviscid fluxes, the fifth-order WENO scheme has been used. For the WENO scheme formulation according to Jiang and Shu [18,19], formally odd order of N = 5 and an 5-point stencil is employed to calculate the inviscid numerical fluxes. Since that the fifthorder WENO scheme has been successfully applied to numerous simulations of various strong compressible flows, we do not present the details in this paper. The central difference scheme of sixth-order is used for the discretization of viscous fluxes. The discretized equations are integrated in time by means of explicit third-order total variation diminishing (TVD) Runge-Kutta algorithm. The time step is determined according to a proper Courant–Friedrichs–Lewy (CFL) number, which is set as 0.5 in this paper.
2.2. Boundary conditions In the case of a mixing layer, the velocity profiles, when scaled in the crosswise direction with a length proportional to the mixing layer width, collapse into a single profile, which is close to a tanh function [20]. Therefore, for the inflow boundary conditions, the mean streamwise inlet velocity distribution is specified as a tanh-type profile, which is the most common model for analyzing mixing layers [21]: 2
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U(0) =
U1 + U2 2
−
U1 − U2 2
tanh
exhibit a Gaussian shape centered on the mixing layer centerline (Y* = 0) where flow turbulent activity is the most intense. The halfwidth of the transverse profiles are in particular accurately reproduced by the simulations. Apart from that, a test case is calculated with the same parameters except for the coarser computational mesh to examine the level of grid dependency. The cell size is about one and a half times larger than that for the fine grid case. It can be seen that compared to the case with coarse grids (CG), the calculation with fine grids (FG) show better agreements with the experimental data. To further confirm the grid independence of the present numerical procedures, comparison of momentum thickness versus X direction in coarse and fine grid levels is presented in Fig. 3. It can be seen that though a slight difference, the results obtained in the two grid sets agree well with each other. Therefore in this paper, the computational mesh with fine grid sets possesses property of high-resolution and can be employed in the present numerical study.
( ) y 2θ (0)
V(0) = 0 W(0) = 0 for three − dim ensional cases
(1)
Where θ (0) is the initial mixing layer momentum thickness at the inlet position. For the expression of U(0), the first term indicates the mean flow. The last term describes a smoothly varying flow within a transition layer with width 2θ (0) . To excite the flow instability and seed the laminar breakdown of the mixing layer, the flow is excited at the upstream boundary of the calculation domain. The disturbance function of transverse-velocity fluctuations is set as:
f1′ = AΔUG (y) sin(2πfv t + φ)
(2)
Here, A is the amplitude, G(y) is Gaussian function, φ is the random phase ranging from 0 to 2π, fv is the most unstable frequency of the mixing layer, ΔU is velocity difference and ΔU = U1 − U2 , t is the time. Apart from that, for three-dimensional cases, to obtain natural happened mixing layers, the supersonic mixing layer is periodically excited by a superimposed three-dimensional oblique waves with the following form,
2.4. Precision estimates For large scale simulations of complex compressible flows, estimating precision and errors accumulation is necessary [26]. Errors accumulation takes place for successive time steps, and it should be evaluated for each simulation case. Here we employed the method used by Smirnov et al. [27] to evaluate the error accumulation and characterize the reliability of the simulation results. Detailed description of the method can be found in Ref. [27]. Table 2 presents the accumulation of error for different grid resolutions and physical time simulations using the present numerical procedures. 100000 time steps are calculated for both cases to ensure the accuracy of the turbulent statistical results. It can be seen that for coarse grid case, the accumulation of errors takes place in a slightly faster manner, while decreases with the increase of grid resolution. Besides, the results demonstrate that the present fine grid set and numerical procedures can provide high reliability for the study.
f2′ = B(y) (cos(± βz + ωt ) + cos( ± βz + 0.5ωt ) + cos( ± βz + 0.25ωt )) (3) Here, B(y) is Gaussian Function, ω corresponds to the most unstable frequency obtained from the linear theory proposed by Michalke [21], and ω = 2πfvt. Sandham et al. [22] have demonstrated that simulations with forcing of linear instability waves can produce the development of large-scale structures similar to a fully nonlinear computation with a random initial condition. β is the wave numbers in spanwise direction, and its value can be obtained in Sandham et al.[ 22]. The supersonic outflow boundary condition is imposed at the transverse boundaries in Y direction, since the reflection waves are not expected to reflect back to affect the calculating main flows. At the outlet of the calculating domain, the boundary condition is also set as outflow boundary condition, which has been discussed and demonstrated in former research [23]. Besides, for three-dimensional cases, the periodical boundary condition is enforced in spanwise (Z) directions. The detailed three-dimensional computational model is presented in Fig. 1.
3. Results and discussions 3.1. Visualization and analysis of flow structures Since it has been demonstrated by Martha et al. [28] that using isosurface of vortex intense is a good way to visualize the structures in the flow, we first employed Q-criterion to detect the vortex structures and analyze the mixing characteristics of supersonic mixing layer. Fig. 4 depicts the three-dimensional visualization of supersonic spatially developing mixing layer with Mc = 1.0. The variable Q is defined as the second invariant of the velocity gradient tensor and it is well suited to vortex-field identification [29]. The iso-surface of the vortex structures with a non-dimensional Q value of 0.0005 is presented in Fig. 4. The color changes from red to blue as Y increases. It can be clearly seen that large-scale structures and the evolution process are well captured through the present numerical scheme. After a period of shear action of upper and lower streams near the inlet, typical K-H
2.3. Grid specification and numerical validation The grids in the mixing layer area are highly refined and the corresponding shear layer vortex structures are well resolved. In Y direction, the grids are clustered towards the centerline to provide a proper resolution of the mixing layer. For two-dimensional cases, the grid points in X direction are uniform spaced, and for three-dimensional cases, in Z directions the grid points are uniform spaced as well. The computational domain size is defined as Lx × Ly × Lz = 240 × 60 × 15 mm3 with the fine grid sets of Nx × Ny × Nz = 1440 × 240 × 90. The minimum grid space ΔXmin, ΔYmin, ΔZmin is 0.167 mm, 0.08 mm and 0.167 mm, respectively. The Reynolds number based on the initial vorticity thickness is about Re(δω) = 400. To validate the present numerical schemes, the comparison between the numerical results and former experimental and numerical data are conducted. We carried out the numerical simulation of a classical compressible mixing layer model, which has been experimentally investigated by Goebel and Dutton [16]. The flow parameters are listed in Table 1. Fig. 2 presents the comparison of distribution of mean velocity and streamwise turbulent fluctuations compared with the experimental data and some recent DNS results [24,25]. It can be seen that in self-similar region of the flow, the agreement between numerical and experimental data is good. Moreover, for velocity fluctuations, the simulation profiles
Fig. 1. Schematic of the computational model. 3
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vortex structures namely Λ structures roll up in transition region due to the initial K-H instability. Since it has been demonstrated by Sandham et al. [11] that three-dimensional instability plays a dominate role in the supersonic mixing flow when Mc is larger than 0.6, in the present simulation (Mc = 1.0) after a short distance in transition region, Λ structures have a tendency to develop in transverse direction to form three dimensional structures. Apart from that, another 3-D test case using a coarser grid set is conducted to verify the grid independence of the flow structures prior to transition to turbulence, as shown in Fig. 4(b) it can be found that with coarser grid set the evolution of three-dimensional structures are clearly revealed in the instantaneous visualization result as well, showing that the present fine grid set is suitable for the next step study. To clearly illustrate the evolution properties, Fig. 5 depicts more details of the structures with Q = 0.001 in transition region. Due to three-dimensional instability, the Λ vortices evolve into a new structure type called hairpin structures quickly. The flow regions in the dotted boxes A, B, C and D clearly show the evolution of hairpin vortices in the present mixing layer. The two legs of hairpin structure tend to rotate and distort, and eventually breakdown into small scale slender structures. The head of the hairpin vortices experiences instability and stretches and rises in transverse direction. Since the stretch and breakdown process can improve the transport of momentum and scalar of upper and lower streams [30,31], the development and evolution of hairpin vortex structures play a dominant role in the mixing process of the flow. Besides, it should be noted that the preponderance of hairpinlike structures has recently been reported in DNS of a spatially developing boundary layer by Ringuette et al. [32]. Both their research and our findings indicate that hairpin-structures play an important role in the breakdown of shear flows.
Table 1 The flow parameters of experiments conducted by Goebel and Dutton [16].
Upper stream Lower stream
U/(m/s)
M
ρ/(kg/m3)
P/(kPa)
Mc
519 409
2.04 1.4
1.0 0.76
46 46
0.2
3.2. Effects of density ratios variation To highlight the mixing mechanisms affected by the variation of Rρ, the velocity ratio Ru and convective Mach numbers Mc are set as constant, and the inflow parameters are listed in Table 3. To evaluate the mixing properties of supersonic mixing layers, turbulent statistical analysis is necessary. Fig. 6 depicts the normalized mean streamwise velocity isolines of different density ratios (Rρ = 0.75, 1.0, and 1.5 respectively). The isoline value Ψ is calculated based on the following formulation:
Ψ = (U − U2)/ ΔU
Fig. 2. Distribution of mean velocity and streamwise turbulent intensity.
(4)
With the development of the mixing layer, the distance between the transverse locations where Ψ = 0.1 and Ψ = 0.9 show character of linear increase. Meanwhile, the evolution of mixing layer has a tendency of inclination to the side of which the velocity is lower, this phenomenon is due to that the engulfment of upper and lower streams is asymmetric when the inflow velocity is different, and the reason will be discussed in detail in the following section. To quantitatively analyze the mixing layer growth rate of the flow, the velocity thickness is defined based on the velocity isolines,
δ = YU0.9 − YU0.1
(5)
Here, YU0.9 and YU0.1 denote the transverse coordinates where U = U1 − 0.1ΔU and U = U2 + 0.1ΔU , respectively. The velocity thickness versus X is presented in Fig. 7. It is clearly observed that with the flow downstream the streamwise direction, the influence of the change of density ratio on the velocity thickness growth rate is slight. As Rρ increases, the degree of that thickening is not significant as a whole. Here we concluded that Rρ has no obvious effect on the evolution of velocity thickness. This conclusion is consistent with the observations obtained in the experiments of Brown and Roshko [6]. Whereas, the momentum thickness under various Rρ conditions shows different evolution properties, as presented in Fig. 8. It can be seen that although a linear growth tendency is observed for all the three
Fig. 3. Comparison of momentum thickness evolution with different grid resolutions.
4
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Table 2 Error estimates. Allowable error 5% 5%
Grid resolution 960 × 240 × 90 1440 × 240 × 90
Physical time simulated (ms) 2.5 2.5
Number of time steps 100000 100000
Accumulated error 0.0000000005950405 0.0000000005950401
Allowable number of time steps 10
Reliability 2.6571952·105 2.6571969·105
2.6571952·10 2.6571969·1010
Table 3 The flow parameters with the varying of Rρ.
Ru Mc Rρ
Case 1
Case 2
Case 3
0.6 0.65 0.75
0.6 0.65 1.0
0.6 0.65 1.5
Fig. 4. Instantaneous overall visualization of vortex structures with Mc = 1.0. (a) Fine grid case, (b) Coarse grid case. Fig. 6. The distribution of normalized streamwise mean velocity isolines of different Rρ: (a) Ru = 0.6, Mc = 0.65, Rρ = 0.75, (b) Ru = 0.6, Mc = 0.65, Rρ = 1.0, (c) Ru = 0.6, Mc = 0.65, Rρ = 1.5.
cases, at the condition of Rρ = 1.0, the growth rate of the momentum thickness is slightly higher than that of the two streams with different densities. It is obvious that the influence of density ratio cannot be ignored. It has been demonstrated by Dimokatis et al. [33] that for compressible mixing layers with the same inflow densities, the engulfment and entrainment of the flow possess characteristic of
symmetry, and the symmetry can help to increase the level of the flow intensity, which contributes much to the higher growth rate of momentum thickness in fully turbulent region.
Fig. 5. Detailed view of vortex structures in transition region of Mc = 1.0 with iso-surface of Q = 0.001. 5
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rises with more fluctuations. Whereas, at Ru = 0.6 and 0.75 the flow is more stable, the fluctuations are slight and the thickness increase linearly. 3.4. Effects of convective Mach numbers variation Although a consensus has been reached that the increase of Mc would increase the flow compressibility while decrease the mixing layer growth rate, here to make comparisons, it is still essential to make a detailed and quantitative analysis of the influence of Mc on mixing properties. Table 5 lists the inflow parameters with Rρ and Ru fixed as constant. Fig. 12 presents the normalized streamwise mean velocity isolines (Ψ = 0.1, 0.5, and 0.9) of different Mc. Due to the compressibility effects, with the increase of Mc, the mixing layer experiences longer shear action distance. At the condition of Mc = 1.0, the distance for the shear action is about 80 mm, and the transition of the flow is heavily delayed. Here we concluded that with the increase of Mc, the effect on the suppression of rolling up process of K-H structures contributes much to the decrease of mixing layer growth rate. This conclusion can be quantitatively drawn from the analysis of velocity and momentum thickness evolutions, as shown in Figs. 13 and 14. In the near field where the shear action occurs, the values of δ and θ both decrease monotonically with the increase of Mc, indicating that high compressibility inhibits the mixing layer growth rate through the transition delay of the flow. Apart from that, it should be noted that in the far flow field at X > 140 mm, the fitting linear growth rate for all the three cases are nearly the same when keeping Rρ and Ru constant. Going back to the inconsistent conclusion drawn by Clemens et al. [13] and Debisschop et al. [14], it can be easily understood from the present research that for different Rρ and Ru conditions, the variation of only Mc may lead to different conclusions.
Fig. 7. The distribution of velocity thickness of different Rρ.
3.5. Turbulent intensity analysis The Reynolds stresses in the mixing layer are found to be closely associated with mixing process and the vortex structures. For free shear layers, the flow will enter into a self-similar status in fully developed region, meaning that both mean variables and turbulence Reynolds stresses behave in a self-similar manner [34,35]. Thus in the present section, to reveal the effects of different flow parameters on the mixing process, the distribution of Reynolds stresses < u′v′ > /ΔU 2 at streamwise position of X = 0.95Lx under different Rρ, Ru and Mc are researched. It should be noted to show the property of profiles collapse, the transverse coordinate is not normalized by local mixing layer thickness in the present investigations. Fig. 15 depicts the distribution of Reynolds stresses with different Rρ. Though possessing similar shape, the peak values of Reynolds stresses are apparently larger in the core mixing region when the upper and lower streams have the same density, resulting in more intense of turbulent fluctuations. Larger turbulent fluctuation intensity will drastically promote the flow instability and strengthen the flow three-dimensionality, and more large-scale structures will breakdown into small-scale vortices, which can increase the interface area and increase the mixing level. Therefore, as discussed above in Fig. 8, in self-similar region the mixing layer growth rate at Rρ = 1.0 is slightly larger, which shows well agreement with the Reynolds stresses distributions here.
Fig. 8. The distribution of momentum thickness of different Rρ.
3.3. Effects of velocity ratios variation In this section the variables Rρ and Mc are held to be constant to analyze the effects of Ru on the mixing process. Three cases with different velocity ratios are calculated and Table 4 lists the calculation parameters. Fig. 9 depicts the normalized streamwise mean velocity isolines (Ψ = 0.1, 0.5, and 0.9) of different Ru. With the increase of Ru, the distance between Ψ = 0.1 and 0.9 decrease drastically, indicating that Ru has great influence on the mixing layer growth rate. To better analyze the effects, Fig. 10 presents the velocity thickness versus X of three Ru conditions. It can be revealed clearly that the increase of Ru leads to an obvious reduction of the velocity thickness. Fig. 11 illustrates the observed variation of momentum thickness along the streamwise direction of the flow. At Ru = 0.45 and 0.6, the flow becomes instability quickly, while for the flow at Ru = 0.75, the flow experiences a long distance of shear action (40 mm approximately) and then K-H vortices roll up and the momentum thickness rises at a much lower rate. The results revealed that although with the same Mc, the variation of Ru has important influence on the mixing process. Meanwhile, it can be seen that due to the big velocity difference, the flow possesses much intensity at Ru = 0.45, and the momentum thickness
Table 4 The flow parameters with the varying of Ru.
Ru Mc Rρ
6
Case 1
Case 2
Case 3
0.45 0.65 1.0
0.6 0.65 1.0
0.75 0.65 1.0
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Fig. 11. The distribution of momentum thickness of different Ru. Table 5 The flow parameters with the varying of Mc.
Fig. 9. The distribution of normalized streamwise mean velocity isolines of different Ru: (a) Mc = 0.65, Rρ = 1.0, Ru = 0.45, (b) Mc = 0.65, Rρ = 1.0, Ru = 0.6, (c) Mc = 0.65, Rρ = 1.0, Ru = 0.75.
Rρ Ru Mc
Case 1
Case 2
Case 3
0.75 0. 6 0.4
0.75 0.6 0.65
0.75 0.6 1.0
Fig. 10. The distribution of velocity thickness of different Ru.
Fig. 16 shows the distribution of Reynolds stresses in transverse direction at different Ru conditions. It can be found that in the external part of the mixing layer at Ru = 0.75, the value of Reynolds stress is apparently lower, indicating that the mixing zone where turbulent fluctuation occurs is much smaller than the cases of Ru = 0.46 and Ru = 0.6. The same phenomenon is also appeared in Fig. 17. It can be seen that when the flow is at highly compressible conditions of Mc = 1.0, the fluctuation is heavily inhibited, leading to observed discrepancies between these three cases. Apart from that, as mentioned in section 3.2, it is interesting to point out that the mixing region has the tendency to incline to the incoming stream side with low velocity. Especially with smaller velocity ratios at Ru = 0.45 and Ru = 0.6, the phenomenon is more notable. The results, though first shown in the present paper, have been reported by Konrad et al. [36] in their investigations on incompressible shear layers. The reason is that the entrainment and engulfment of upper and lower streams are not symmetrical due to the velocity difference of the
Fig. 12. The distribution of normalized streamwise mean velocity isolines of different Mc: (a) Rρ = 0.75, Ru = 0.6, Mc = 0.4, (b) Rρ = 0.75, Ru = 0.6, Mc = 0.65, (c) Rρ = 0.75, Ru = 0.6, Mc = 1.0.
streams. To quantitatively analyze the influence of different flow parameters on turbulent statistical properties, Table 6 presents the peak intensities. It can be noted that though slight variation, the effects of Rρ and Ru on the peak intensities are not significant, while with the increase of Mc, the peak values of Reynolds stresses experience sharp decrease.
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Fig. 16. Reynolds stresses distribution of different Ru.
Fig. 13. The distribution of velocity thickness of different Mc.
Fig. 14. The distribution of momentum thickness of different Mc.
Fig. 17. Reynolds stresses distribution of different Mc.
results, the correlation coefficient Cuu in X-Y plane can be defined by the following formula,
Cuu (X0 + ΔX , Y0 + ΔY ) =
u′ (X0 , Y0)⋅u′ (X0 + ΔX , Y0 + ΔY ) u′ (X0 , Y0)2 ⋅ u′ (X0 + ΔX , Y0 + ΔY )2
(6)
Where (X0, Y0) is reference location, u' is the streamwise fluctuation velocity. Poggie et al. [37] have demonstrated that a set of more than 200 instantaneous images can ensure the stability of the calculating results. To analyze the vortex topology characteristics in self-similar turbulent region, the reference location is chosen at (192 mm, 30 mm) far downstream the flow field. Fig. 18 depicts the spatial correlation of flow structures versus Rρ variation. The correlation contours are seen to be approximately elliptical in shape with the major axis aligned with the flow direction, which shows well agreement with former results [37]. Bourdon et al. [38] have demonstrated that the shape of correlation contours represents the mean size of the flow structures. Thus, to quantitatively analyze the vortex topology properties, the least square ellipse fitting method is used to gain the size of the vortices. The region enclosed by contour level of 0.5 has principal axes of d1 = 7.348 mm, 7.932 mm, 7.857 mm and d2 = 3.446 mm, 3.398 mm, 3.721 mm for the cases of Rρ = 0.75, 1.0 and 1.5, respectively. Though possessing a slight larger size for the condition of Rρ = 1.0, the effects of Rρ variation on the size of flow structures are not significant. Whereas, at Rρ = 1.0 the
Fig. 15. Reynolds stresses distribution of different Rρ.
3.6. Vortex topology analysis To investigate the vortex topology characteristics, the spatial correlation analysis method is employed. Based on the present simulation 8
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Table 6 Comparison of peak Reynolds stresses with different inflow parameters. Ru=0.6, Mc=0.65 〈u'v'〉/△U
2
Rρ = 0.75 0.05595
Mc = 0.65, Rρ = 1.0 Rρ = 1.0 0.06539
Rρ = 1.5 0.05658
Ru = 0.45 0.0602
Ru = 0.6 0.06539
Rρ = 0.75, Ru = 0.6 Ru = 0.75 0.05461
Mc = 0.4 0.07277
Mc = 0.65 0.05595
Mc = 1.0 0.04391
Fig. 18. Spatial correlation of flow structures versus Rρ: (a) Ru = 0.6, Mc = 0.65, Rρ = 0.75, (b) Ru = 0.6, Mc = 0.65, Rρ = 1.0, (c) Ru = 0.6, Mc = 0.65, Rρ = 1.5.
Fig. 19. Spatial correlation of flow structures versus Ru: (a) Mc = 0.65, Rρ = 1.0, Ru = 0.45, (b) Mc = 0.65, Rρ = 1.0, Ru = 0.6, (c) Mc = 0.65, Rρ = 1.0, Ru = 0.75.
Fig. 20. Spatial correlation of flow structures versus Mc: (a) Rρ = 1.0, Ru = 0.6, Mc = 0.4, (b) Rρ = 1.0, Ru = 0.6, Mc = 0.65, (c) Rρ = 1.0, Ru = 0.6, Mc = 1.0.
short axis reach the minimum of 5.146 mm and 2.322 mm. Going back to Figs. 10 and 11, it can be seen the structure topology evolution corresponds well to the mixing process of different flow parameters. The smaller sizes of the mean vortex structures are responsible for the inhibition of the mixing growth rate at higher Ru condition. The spatial correlations of flow structures versus Mc are presented in Fig. 20. An interesting phenomenon is that as Mc increases, the
shape of the mean structure has the tendency to be compressed and become narrow, since the ratio of long axis and short axis is 2.334, which is much larger than that of the other two cases. Fig. 19 shows the correlation plots for the cases with different Ru. It can be obviously seen that with the increase of velocity ratios, though having a shape similar to an ellipse, the size of mean flow structures experiences sharp decrease. When Ru is fixed to 0.75, the long axis and 9
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influences on the size of flow structures are not significant, while the shape of the mean structure tend to be compressed and become narrow at Rρ = 1.0 condition. For Ru variation, the size of mean flow structures experiences sharp decrease with the increase of Ru, and the smaller size of the mean vortex structures are responsible for the inhibition of the mixing growth rate at higher Ru condition. As Mc increases, the vortex size experiences decrease but not significant. The present investigations could provide explanations for the inconsistent conclusions drawn by former researchers during their experimental and numerical works.
vortex size experiences decrease but not significant. Going back to Figs. 13 and 14, it can be found that the phenomenon is reasonable since that in far flow field region the fitting linear growth rates for the three different cases are nearly the same when keeping Rρ and Ru constant. Therefore, although Mc has been demonstrated to be an important non-dimensional parameter to evaluate the flow compressibility, the varying of only Mc usually leads to inconsistent conclusions in previous investigations. The effects of each of the three main flow parameters, including Rρ, Ru, and Mc, on mixing process and vortex topology evolution are different, and it is necessary to analyze the influence of each parameter respectively.
Acknowledgments 4. Conclusions and remarks The authors would like to express their thanks for the support from National Natural Science Foundation of China (Grant Nos. 11272351, and 91441121), and the Graduate Student Research Innovation Project of Hunan Province (Grant No. CX2016B001).
In the present paper, direct numerical simulations of spatially developing supersonic planar mixing layers are conducted with focusing on the effects of varying of three main flow parameters including Rρ, Ru and Mc on the mixing process and vortex topology properties of the flow. After validation of the numerical procedures, two indexes, namely velocity thickness and momentum thickness are employed to evaluate the mixing characteristics with different inflow parameters. The distribution of Reynolds stresses are analyzed to show the turbulent intensities. The spatial correlation analysis is carried out to investigate the vortex topology properties influenced by different flow parameters. The following conclusions can be drawn based on the results:
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(1) The comparison between present numerical results and former experimental and numerical data reaches well agreements, indicating that the present numerical method could efficiently capture the characteristics of compressible mixing layer flow. The profiles show a Gaussian shape centered on the mixing layer centerline where flow turbulent activity is the most intense. The half-width of the transverse profiles are in particular accurately reproduced by the simulations. (2) The overall development and evolution of three-dimensional vortex structures in supersonic planar mixing layers are well reproduced. The mixing layer experiences shear action region, turbulent transition region and fully turbulent region downstream the flow field. In transition region, typical flow structures, including Λ vortices, hairpin structures and braid vortices are well captured, and the role of hairpin vortices played in mixing process of shear layers is analyzed in detail. (3) The effect of Rρ variation on the velocity thickness growth rate is not significant, while the momentum thickness growth rate becomes slightly higher when the inflows have the same density. With the increase of Ru, the indexes evaluating mixing efficiency including velocity and momentum thickness are both heavily depressed. However, for the variation of Mc, the mixing layer experiences longer shear action distance at Mc = 1.0 and the transition of the flow is heavily delayed, which contributes much to the low growth rate in the near flow field. Besides, in the far flow field at X > 140 mm, the fitting linear growth rate for all the three cases are nearly the same when keeping Rρ and Ru constant. (4) The effects of each of the three main flow parameters on the Reynolds stresses are different. The peak values of Reynolds stresses in self-similar region experience sharp decrease with the increase of Mc. Whereas, for the cases of varying of Rρ and Ru, the change of peak values of Reynolds stresses is slight. The smaller Reynolds stresses in transverse region, the less intense of turbulent fluctuations, which is responsible for the inhibition of mixing process. The mixing region has the tendency to incline to the incoming stream side with low velocity. The results are first shown in the present paper and the reasons are analyzed. (5) The spatial correlation analysis results show that the effects of each of the three main flow parameters on vortex topology lead to different mean structure sizes and shapes. For Rρ variation, the 10
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