Finally, a comparison is made to the models mentioned in Section 3.2.3. To clarify, these are the models by Mikaelian and Charakhch’yan. These models are:
Mikaelian’s reshock model:
dh2
dt = C∆V2A
+
(10)
WithC=0.28, as determined by Mikaelian andC=0.56, as determined by Ukai et al[2]. And Charakhch’yan’s model:
dh2
dt = β∆V2− dh1
dt (11)
With β = 1.25, as determined by Charakhch’yan and β = 0.68, as determined by Ukai et
al.[2]
One thing that can be noted immediately before the results are viewed, is that neither model shows any dependence on initial conditions like wavelength or initial amplitude. This is in contrast to the findings in Section 5.4, where a dependence of MZ growth rate on both initial amplitude and wavelength was found. For the results both models are considered, as well as a newly proposed constant values by Ukai et al.[2] for both models. The results are shown below and show the amplitude growth for different situations, and the predicted growth rate for each model. Since post-reshock growth rate is the same for different reshock times and the same as the base case, the results of these experiments are not compared to the models. Firstly the base case is observed. The results of the post-reshock growth rate for the base (40-1) case and growth rate predictions are shown in Figure 25.
Figure 25: Post-reshock growth rate for the base (40-1) case and predicted growth rate for different models
In this figure, Charakhch’yan’s model seems to match almost perfectly with the initial growth rate after reshock and all other models underestimate the post-reshock growth rate. It can also be noted that there is a relatively large difference in growth rate between both models using their originally suggested coefficients (a factor 3). Afterwards, the results for the other simulations are considered. They can be found in Figure 26.
Figure 26: Post-reshock growth rate for different wavelengths and initial amplitudes and predictions by different models
From the figures of the different simulations it can be seen that the difference in post-reshock growth rate is indeed significant compared to the predicted growth rates by the models. The early time growth rate of the 40-0.67 case matches Charackhk’yans model, however, for all other cases, Charakhch’yan’s model overestimates the growth rate. Lower values for the
λ/a0ratio result in smaller post-reshock growth rates, which is not reflected in the models. for large wavelengths or small amplitudes, the models predicting a smaller growth rate are more correct, but no single model can be considered as correct for all cases.
The reason for a lack of correct predictions by all models is that they ignore the dependency on the initial conditions. A possible solution could be to find a constantC(λ,a0)orβ(λ,a0), dependent on the initial conditions, that could correctly predict the post-reshock growth rate, however this is outside of the scope of the current research. In the work by Ukai et al.[2] it is noted that the proposed value ofβby Charakhch’yan can be determined with the
equation πhr
λ , withhr the mixing length at reshock. However, this relationship is denied by
6
Conclusions & Recommendations
In the present research, a single-mode gaseous interface under reshock was examined. Sim- ulations were performed that mimic a real-life shock-tube situation. The main research fo- cused on finding the effects of various parameters on the mixing zone growth rate after reshock. The examined parameters are: Reshock time, wavelength, initial amplitude and Mach number. In the problem statement a hypothesis was provided:Post-reshock MZ growth is weakly dependent on the initial conditions, such as initial wavelength and amplitude. The re- sults of the research prove this hypothesis right. The results of simulations proved that a relationship exists between these parameters and the MZ growth rate, where a larger wave- length resulted in a smaller/slower mixing zone growth rate. The use of a smaller initial amplitude also resulted in a similar effect. However, where the difference growth rate of the different wavelength cases was constant in the dimensionless domain (and therefore predi- cable) for all (single-mode upon reshock) wavelength cases, the different amplitudes show variation in the dimensionless domain. The result of the different reshock times in the sim- ulation showed no effect of reshock time on the post-reshock early-time growth rate for any case. The research was not able to convincingly determine the effect of different initial Mach numbers on reshock, due to a lack of usable results. This was caused by the fact that the performed simulations developed into multi-mode interfaces upon reshock and developed turbulent mixing behaviour, which the currently used simulation tool can not simulate accu- rately. The results of the simulations were used to determine the development of the higher order harmonics in the interface. Only a small effect on the 2nd harmonic in the interface was found, where larger wavelengths delayed the growth of the 2nd harmonic after reshock and larger initial amplitudes result in higher growth rates of the 2nd harmonic after reshock. Finally the reshock models by Mikaelian and Charakhch’yan were investigated to verify if either provides an accurate prediction of post-reshock growth rates. No model was found to match the post-reshock growth rate of all cases. This was attributed to the fact that these models do not take into account the dependency on the initial conditions that was examined in this research.
The current research indicates a previously ignored dependency on initial conditions. Ex- ploring and mapping this dependency can help researchers and engineers by being able to perform mixing experiments that involve a reshocked gaseous interface in a more controlled manner. For instance by using initial conditions that provide a slower post-reshock MZ growth rate. This can be useful for examining reshocked RMI as well as for controlling the mixing rate in any application of the reshocked RMI gas mixing process.
In the current research the effect of different initial Mach numbers was investigated, but no definite findings could be determined due to a lack of usable simulation research. It is sug- gested to continue to research into the effect of this parameter on the reshock growth rate development. Further research can be conducted into finding a model that can correctly pre- dict the post-reshock growth rate for varying initial conditions. This can be done by finding a constant in the presently used models that is dependent on the initial conditions, or by cre- ating a completely new model that takes these dependencies into account. Creating a model that can accurately predict post-reshock MZ growth rates can be helpful in future research and experiments. Finally, since the current research was limited to 2D, single-mode, and non-turbulent interfaces upon reshock much more research can be done into 3D situations and multi-mode interfaces including turbulent mixing behaviour.
7
Acknowledgements
I’d like to thank the following people for making this report possible: prof. Xisheng Luo for providing the assignment and making it possible to come USTC and China to perform the internship. Dr. Juchun Ding for all support during the assignment as well as the process of coming to China and USTC for the internship. Prof. Harry Hoeijmakers for introducing me to the people at USTC and providing me with the chance to do the internship at USTC. And, finally, my friends Li Ming, Guo Xu, Zelai Xu and Zhangbo Zhou for all the hours they spent explaining various software programs, finding bugs in code and emotional support over the coarse of the internship.
References
[1] T. Si, Z. Zhai, J. Yang, and X. Luo. Experimental investigation of reshocked spherical gas interfaces. Phys. Fluids 24, May 2012. 054101 (2012); doi: 10.1063/1.4711866. [2] S. Ukai, K. Balakrishnan, and S Menon. Growth rate predictions of single- and multi-
mode richtmyer–meshkov instability with reshock. Shock Waves 21, August 2011. DOI 10.1007/s00193-011-0332-0.
[3] M. Brouillette. The richtmyer-meshkov instability. Annual Reviews Fluid Mech. 34, 2002.
[4] L. Liu, Y. Liang, J. Ding, N. Liu, and X. Luo. An elaborate experiment on the single-mode richtmyer–meshkov instability. J. Fluid Mech. vol.853, R2, July 2018. doi:10.1017/jfm.2018.628.
[5] C.C. Long, V.V. Krivets, J.A. Greenough, and J.W. Jacobs. Shock tube experiments and numerical simulation of the single-mode, threedimensional richtmyer–meshkov insta- bility. Physics of Fluids 21, November 2009. 114104 (2009); doi: 10.1063/1.3263705. [6] M. Latini, O. Schilling, and W.S. Don. High-resolution simulations and modeling of
reshocked single-mode richtmyer-meshkov instability: Comparison to experimental data and to amplitude growth model predictions. Phys. Fluids 19, February 2007. 024104 (2007); doi: 10.1063/1.2472508.
[7] E. Leinov, G. Malamud, Y. Elbaz, L.A. Levin, G. Ben-Dor, D. Shvarts, and O. Sadot. Ex- perimental and numerical investigation of the richtmyer–meshkov instability under re- shock conditions. J. Fluid Mech. (2009) vol. 626, 2009. doi:10.1017/S0022112009005904. [8] G. Malamud, E. Leinov, O. Sadot, Y. Elbaz, G Ben-Dorm, and D. Shvarts. Reshocked
richtmyer-meshkov instability: Numerical study and modeling of random multi-mode experiments. Physics of Fluids 26, August 2014. 084107 (2014); doi: 10.1063/1.4893678. [9] Tecplot.com. Tecplot, data visualization & analysis. Tecplot website, 2019. https:
//www.tecplot.com/.
[10] National Aeronautics and Space Administration (NASA). Schlieren flow visualization. NASA website, 2019. https://www.grc.nasa.gov/www/k-12/airplane/tunvschlrn. html.
[11] Juchin Ding, Ting Si, Mojun Chen, Zhigang Zai, Xiyun Lu, and Xisheng Luo. On the interaction of a planar shock with a three-dimensional light gas cylinder. J. Fluid Mech. (2017), vol. 828, pp. 289–317. Cambridge University Press 2017, July 2017. doi:10.1017/jfm.2017.528.
[12] M. Latini, O. Schilling, and W.S. Don. Physics of reshock and mixing in single-mode richtmyer-meshkov instability. Physical review E 77, August 2007. 026319 2007.
[13] O. Schilling and M. Latini. High-order weno simulations of three-dimensional reshocked richtmyer-meshkov instability to late times: Dynamics, dependence on initial conditions, and comparisons to experimental data. Acta Mathematica Scientia 30B(2), 2010. DOI 10.1007/s00193-011-0332-0.
A
Paper reviews
In these reviews, all figures and information are from the papers mentioned. A.1 Growth rate predictions of single- and multi-mode
Richtmyer–Meshkov instability with reshock S. Ukai, K. balakrishnan, S. Menon[2]
In This paper, a numerical study is done for Richtmyer-Meshkov instability(RMI) in four different cases: single- mode 2D and 3D and multi-mode 2D and 3D situations. Shock-tube experiments are compared to the numerical analysis. A parametric analysis is done to find the effects of initial conditions on late-time growth patterns.
To find an expression for the change of growth rate, several models were used in this paper. The models can be used in single-mode or multi-mode situations. Models applicable in single-mode situations are:
Mikaelian’s potential model:
dh2 dt = 2πhr λ ∆V2A− dh1 dt (12) Charakhch’yan’s model: dh2 dt =β∆V2A− dh1 dt (13)
The second type of model, are the models applicable for multi-mode situations, in this case: Mikaelian’s reshock model:
dh2
dt = C∆V2A
+
(14) These models all agree with the assumption that growth rate is not a function of mixing length at reshock (hr) or wave number (k) before the second shock. The goal of the paper
is to determine if this is indeed the case. However, Charakhch’yan’s and Mikaelians mod- els disagree for small dh1
dt . The reason for this can be how the mixing zone, is determined.
Because a diffusion layer is present in most physical situations, the definition of the mixing zone is therefore determined by how much of the diffusion layer is is incorporated in the definition of the mixing zone. Charakhch’yan’s model does not consider the diffusion layer and turbulent mixing that is present in physical experiments. Because there is a scarcity of data on single mode RMI after reshock, Charakhch’yan’s model is not validated and the co- efficient is re-investigated in the paper.
Because it is hard to predict the initial conditions of the perturbation shapes, several solu- tions were sought, but none of them offered an exact prediction of the initial perturbation shape. Therefore it is hard to compare numerical results to empirical data.
Finally another problem for validation of the data is that only experimental data for 3D multi-mode RMI is available. For this paper it was chosen to use the data of reshocked RMI experiments of models by Leinov et al.
Simulation methodology
diffusion layer exists that makes it difficult to define the exact interface. To define the inter- face two methods can be used:
The threshold method, where the interface is defined by the average mass fractionYif this becomes lower than a certain amountεit defines the interface:
ε <(Yair)<1−ε (15)
This method does mean that the mixing length is influenced by the choice ofε
The other method is to measure the iso-contour of the mass fraction, this works only for single-mode studies.
The difference in definition is shown below in Figure 27. In this paper the threshold method is used.
Results
Figure 27: Difference in mixing layer estimate by different methods
The results point to several more problems concerning the reshocked RMI analysis. It also gives interesting points that can be researched further
First of all, during the grid sensitivity check, it was found that, after the expansion wave hits the interface in 2D and 3D single-mode cases, it becomes grid dependent, as a result the growth rate becomes grid dependent.
Furthermore, validation of multi-mode RMI is hard to perform, due to a high dependency of on the statistical approach,
a parametric study is done for single- and multi-mode RMI. It is found that for single-mode RMI:
• Mikaelian’s potential model is not suitable for large perturbation amplitudes. • ∆V2is predicted to be linearly correlated to growth rates.
• Mikaelian’s potential model over-predicts the growth rates by comparingβtot 2πλhr
• Differences exist between the 2D and 3D flows in Mikaelian’s reshock model, where the C values differ from each other and from the originally proposed value.
Proposed explanations for these differences are:
1) In 3D the vortex stretching term of the inviscid compressible vorticity relation is activated, this can possible strengthen vorticity if the vortex rings are stretched.
2) The geometry of perturbation is different for 2D and 3D cases. In 3D the RMI has a point contact to the shock at bubble and spike fronts, while a 2D situation has a line contact to the shock. This can cause larger vorticities of fronts that cause larger growth rates.
For the single-mode RMI analysis, no correlation is found between post-reshock growth rate and wavelength and mixing length at reshock, a linear relation is found with∆V2. However, there is a 1.6 times larger growth rate in 3D than in 2D.
For the multi-mode case the growth rate is expected to depend on randomness of the initial perturbation. Due to this randomness two sets of solutions were found:
The ’rapid growth’ solution was for largera0andkmax. with growth rates comparable to the
single-mode RMI.
The ’slow growth’ solution was found for largera0andkmax. in this case the foundCvalue
was only 40% of that of the rapid growth case.
To define the range of these cases a randomness factor, assuming it is only dependent ona andk, was suggested:
R= a0kmax Ly
(16) Where, according to empirical research, rapid growth in 2D is likely forR≤ 0.4 and for 3D ifR ≤ 0.2. Finally it is important to define the slow growth and rapid growth in a range of C. Though expirements this was determined to be: 0.33< C < 0.44 which agree well with the data determined in the numerical simulation.
Finally, the growth rate is affected by the randomness: if the reshock hits a well mixed in- terface (high randomness), this will decrease the growth rate, but increase mixing. If the reshock hits an interface with a very low randomness, the beheviour follows single-mode RMI behaviour.
A.2 Experimental and numerical investigation of the Richtmyer– Meshkov instability under re-shock conditions
E. Leinov, G. Malamud, Y. Elbaz, L.A. Levin, G. Ben-Dor, D. Shvarts, O. Sadot[7]
In this paper an experimental and numerical study about RMI induced mixing under reshock is done. An experiment is carried out in a shock-tube, where the effects of changing the mix- ing zone (MZ) width at reshock, changing the initial shock Mach number and the reshock Mach number on MZ growth rate are tested. This experiment is done in a shock tube with a moveable end wall to control MZ width at reshock and an elastomeric foam behind the end wall to control reshock strength. the These results are compared to a numerical simulation. This numerical simulation uses a random initial perturbation. The comparison of the exper- imental data with the numerical simulation indicate that the linear growth rate of the of the
mixing zone is a result of the bubble competition process.
The Richtmyer-Meshkov instability is governed by three physical elements: Single bubble evolution, single spike evolution and interaction between neighboring bubbles (bubble com- petition).
bubble competition can happen before and after the reshock. In the bubble competition pro- cess large bubbles overtake the volume originally occupied by smaller bubbles. Layzer(1995) described the bubble behaviour using a simple potential flow model.
The paper mentions several problems regarding the experiment that is to be carried out. For instance, that some experiments find much smaller values for the MZ than others, due to the presence of a wall jet in some experiments. Because of this, a distinction has to be made between wall jet width and MZ width. It was proposed to model the evolution of a single- mode interface accelerated by multiple shock waves by applying the impulsive solution for each shock interaction, forNshock interactions:
∂h ∂t N = N−1