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

  • Aditi SenguptaEmail author
  • Tapan K Sengupta
  • Soumyo Sengupta
  • Vidyadhar Mudkavi
Conference paper
  • 505 Downloads
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 135)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Upon removal of the partition, RTI ensues by linear and nonlinear mechanisms with vortices created by the baroclinic source term (\(\nabla p \times \nabla \rho / \rho ^2\)) triggered by background disturbances. The schematic with boundary conditions are shown in Fig. 1. Onset of instability occurs at very high wavenumbers at the junctions of the interface with sidewalls, which propagates along the interface [9]. Thus, the disturbance migrates from very small to larger scales. It has been reasoned [14] that if one uses a diffusive/dissipative numerical method, then these small scales are attenuated and the process of RTI is not captured. This has prompted many researchers [2, 3, 7, 10, 16] to perform 3D simulations with large scale excitation at the interface, hoping that the smaller scales would be created via vortex stretching subsequently via direct cascade. Such external forcing was also necessary for 3D simulations as the variables were treated as periodic in each horizontal planes.
Fig. 1

Physical and computational domain of RTI problem solved

2 Validation of RTI with Experiment

Present non-periodic simulations reproduce all the experimental features [9], as compared in Fig. 2. Details of numerical methods and auxiliary conditions are described in [14]. In the figure, density contours are shown for early times to trace the RTI onset process. Even at later times (\(t = 12\), 21 and 34) the match between experiment and computation is excellent. One notes the origin of disturbances at the side-walls and Subsequently, very small scale disturbances all along the interface. Such excellent match with experiment suggests superiority of non-periodic over periodic formulation.
Fig. 2

Comparison of computed (left) solution with experimental results of [9] for RTI problem

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.

While contamination from side walls continue after the onset, in the interior also the disturbances grow with structures known as spikes and bubbles. These are shown in Fig. 3 for both the CFL number cases, with the lower CFL number case (in frame (a)) showing slower growth of forming spikes and bubbles. It is important to note that both these elements are heavier than surrounding fluids, yet the one in top is termed as bubble. With passage of time nonlinear growth helps formation of vortices which promote mixing.
Fig. 3

Spikes and bubbles formation during RTI by numerical solution using CFL number of 0.009 (left) and 0.09 (right)

3 Error Dynamics and Spectral Analysis

To understand the numerical method for computing RTI, we adopt the 1D wave equation
$$\begin{aligned} \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0 \end{aligned}$$
(1)
One can express the unknown in a hybrid representation as \(u(x,t) = \int U(k,t) e^{ikx} dk\), with a numerical amplification factor defined as, \(G = U(k,t + \varDelta t)/U(k,t)\). If the initial condition is given by, \(u(x, t=0) = \int A_0 (k) e^{ikx} dk\), then the numerical solution can be expressed as \(u_N (x,t) = \int A_0 (k) |G|^{t/\varDelta t} e^{ik(x - c_N t)} dk\), where the numerical phase speed (\(c_N\)) is 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
$$\frac{\partial e}{\partial t} + c \frac{\partial e}{\partial x} = \left( 1 - \frac{c_N}{c}\right) \frac{\partial u_N}{\partial x} - \int \int \frac{dc_N}{dk} \biggl [ ik' A_0 |G|^{t/\varDelta t} e^{ik' (x -c_N t)} dk'\biggr ] dk$$
$$\begin{aligned} - \int \frac{Ln |G|}{\varDelta t} A_0 |G|^{t/\varDelta t} A_0 |G|^{t/\varDelta t} e^{ik(x -c_N t)} dk \end{aligned}$$
(2)
The second term on the right hand side can be further simplified as \(\frac{dc_N}{dk} = (V_{gN} - c_N)/k\), with the numerical group velocity given by, \(V_{gN} = d\omega _N/dk\). Thus, the error dynamics for convection dominated problems are given by the error metrics: |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.
These properties are shown in Fig. 4, for \(N_c\) from 0.001 to 0.009 with higher digit precision. In frame (a), the quantity \((|G|-1)\) is shown. One notices that this retains a numerical zero value for \(N_c = 0.007\). There is mild instability in an intermediate range of 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 \(|G| \le 1\).
Fig. 4

Numerical properties of used method for solving 1D convection equation: Numerical amplification factor (top), normalized numerical phase speed and numerical group velocity (bottom)

Fig. 5

Computed RTI solutions using CFL no. of 0.09 (left) and 0.009 (right)

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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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Aditi Sengupta
    • 1
    Email author
  • Tapan K Sengupta
    • 2
  • Soumyo Sengupta
    • 3
  • Vidyadhar Mudkavi
    • 4
  1. 1.Department of EngineeringUniversity of CambridgeCambridgeUK
  2. 2.HPCL, IIT KanpurKanpurIndia
  3. 3.Department of Mechanical and Aerospace EngineeringOSUColumbusUSA
  4. 4.Head, CTFD Div. CSIR-NALBangaloreIndia

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