# Effects of Error on the Onset and Evolution of Rayleigh–Taylor Instability

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## Abstract

Here we investigate effects of error in simulating Rayleigh Taylor instability (RTI). The error metrics are evaluated based on the correct spectral analysis of a model equation by Sengupta et al. (J Comput Phys 226:1211–1218, 2007) [13]. The geometry for RTI consists of a rectangular box with a partition at mid-height separating two volumes of air kept at a temperature difference of 70 K. This helps avoiding Boussinesq approximation and the present time-accurate computations for compressible Navier–Stokes equation (NSE) in 2D are reported. Computations for CFL numbers of 0.09 and 0.009 shows completely different onset of RTI, while the terminal mixed stage appears similar. The difference is traced to very insignificant difference in the value of numerical amplification factor for the two CFL numbers.

## Keywords

Numerical Amplification Factor Compressible NSE Error Metrics Numerical Group Velocity Numerical Phase Speed## 1 Introduction

RTI arises at the interface of fluids of different densities, with the heavier one resting above the lighter fluid [8, 11, 15]. We study RTI, where initially air with two different constant temperatures (and hence density) is placed across a non-heat conducting interface, inside a rectangular 2D insulated box. Using same fluid alleviates the problem of tracking the interface. Large discontinuous jump in temperature (\(\varDelta T = 70K\)) will not allow one to apply Boussinesq approximation [6]. Most of RTI simulations [2, 3, 5] are reported using incompressible formulation. Present use of compressible NSE [1, 4] helps remove this major discrepancy. It is noted [6] that *the issue of compressibility is highly challenging and direct numerical simulations are the best way to assess the validity of Boussinesq approximation.* Thus the time evolution of the RTI is traced here by high resolution dispersion relation preserving (DRP) compact scheme for DNS.

## 2 Validation of RTI with Experiment

One of the motivations of the present work originated from the observation [2] that *the availability of even more powerful computers has led to a somewhat ironic state of affairs, in that agreement between simulations and experiments is worse today than it was several decades ago.* To understand this problem, we performed two simulations, with CFL number reduced from 0.09 to 0.009. These exercises will help us understand why results deteriorate with refined calculation.

## 3 Error Dynamics and Spectral Analysis

*k*dependent and not the constant physical phase speed,

*c*. Thus, the numerical dispersion relation is given by \(\omega _N = c_N k\), instead of the physical dispersion relation, \(\omega = c k\), where \(\omega \) and \(\omega _N\) are physical and numerical circular frequency, respectively. Details of this analysis is available in [12, 13]. If we define error by \(e(x,t) = u(x,t) - u_N (x,t)\), then the error propagation equation is given by

*G*|, \(c_N/c\) and \(V_{gN}/c\), as functions of nondimensional number

*kh*(with

*h*as the uniform spacing) and the CFL number (\(N_c = c \varDelta t / h\)). For least error, one must have a neutrally stable method (\(|G| \simeq 1\)) which is dispersion error free (i.e., \(v_{gN}/c \simeq 1\) and \(c_N/c \simeq 1\)) for combinations of \(N_c\) and

*kh*values, near the origin in \((kh, N_c)\)-plane.

*k*for \(N_c =0.008\) and 0.009. This explains, why we do not need extrinsic excitation to trigger RTI. For \(N_c = 0.09\) case, the ensuing instability will be even stronger across wider range of

*k*’s, as evident in Fig. 3. In frames (b) and (c) of Fig. 4, one does not notice any discernible difference of dispersion properties for the range of \(N_c\) shown. This also establishes that |

*G*| make all the differences between the two \(N_c\) values differing by a factor of ten. We note that in [2], the value of \(N_c\) is larger than 0.09, yet the authors required external large scale forcing, as the used filters and sub-grid scale models employed attenuated disturbances with \(|G| \le 1\).

## 4 Summary and Conclusion

One of the interesting aspects of nonlinear stage of RTI and resultant mixing near the interface is the final state of the system. For both the \(N_c\) cases, the density contours are shown in Fig. 5 at later times. It is evident that the vortices at the interface promote mixing and in the long run, flows look similar. The width of the mixing region though evolves at different rates. This inference does not imply that 2D non-periodic flow and 3D periodic flow (in the horizontal plane) would be eventually same. It is due to different dynamics of periodic and non-periodic flows, which causes the energy to appear from the opposite end of the spectrum and the migration of energy across different scales proceed exactly in opposite direction, so much so that the comments in [2] is now understood in its true spirit. The recorded failure in [2] is also compounded by overtly dissipative numerical method, which forced the authors to apply external forcing at the interface at the large scale and expect vortex stretching to take the flow in the right direction via direct cascade mechanism.

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