Instability and acoustic fields of the Rankine vortex as seen from long-term calculations with the tenth-order multioperators-based scheme

Abstract

The instability of the compressible Rankine vortex is investigated using a multioperators-based scheme with the tenth-order approximation to the convection terms of the fluid dynamics equations supplied by the ninth-order dissipation mechanism. The scheme was optimized for high resolution of small scales by choosing its free parameters. The results of long-term calculations with several meshes allowed to reproduce the general scenario of the instability development with self-excitations of different modes and the final stage of rotating quadrupole sound radiation. The calculated frequencies and sound pressure levels are presented for several Mach numbers.

Introduction

The multioperators-based prescribed-order schemes for fluid dynamics proposed in [20], [21] and described in [22], [23], [25], [26] provide a good opportunity to perform long-term calculations with high resolution of small details of flows described by the Euler or the Navier–Stokes equations. In particular, they fit neatly into the requirements of DNS, LES and computational aeroacoustics.

Using multioperators-based schemes, some shear layers instability problems were numerically simulated. The direct numerical simulation of 2D turbulence was reported in [16]. The unsteady behavior of hot subsonic jets generating vortex rings with self-sustained oscillations was investigated in [15]. Very clear pictures of screech waves generated by slightly supersonic underexpanded jets were presented in [27]. The previous experience has shown that the multioperators technique allows one to describe fine flow details using comparatively modest numbers of grid points.

In the present study, we continue to use very high order and high resolution schemes for investigating sound generation due to flow instabilities. We consider now the development of the instability of the Rankine vortex (that is the vortex with constant vorticity inside a circle and the zero one outside it) in the framework of the Euler equations. It is well known that the flow field of the vortex is the exact steady state solutions of both incompressible and compressible Euler equations. However the solution is not stable in the latter case and the Rankine vortex is an oscillatory system emitting sound. There are numerous publications considering its instability, the natural method of attack being the linear theory (see [2], [3], [9]). The Rankine vortex has drawn considerable attention in the context of sound scattering problems [1], [4], [6], [7], [9], [10], [11] in the theory of sound interaction with low Mach number flows.

In the early work [9], the vorticity boundary was initially ($t=0$) disturbed in the form

$$r=a+\epsilon cosn\phi ,$$

where $(r,\phi )$ are the polar coordinates, $a$ is the vortex radius, $\epsilon \ll a$ is a small parameter while the integer $n$ is the mode number. It was shown by Kelvin that in the incompressible case one has for $t>0$ the stable solution

$$r=a+\epsilon cos(n\phi -{\omega }_{n}t){\omega }_n=\Omega (n-1)∕2,$$

where $\Omega$ is a constant vorticity inside the circle. There is no sound radiation in that case. However the solution is unstable if the compressible Euler equations are considered.

In [2], [3], [11], a general form of the compressible Rankine vortex perturbations was used to describe the disturbed flow fields. The resulting eigenvalue problem solved numerically in [2], [3] produced modified frequencies ${\omega }_n^{\prime }$ and the increments ${\delta }_n$ of the unstable $n$th modes for various Mach numbers.

It was found in particular [9] that the increment ${\delta }_2$ of exponential growth of the most unstable mode $n=2$ is

$${\delta }_2=\frac{\pi {\mathit{M}}^4\Omega }{64}$$

where $\mathit{M}$ is the Mach number defined by

$$\mathit{M}=\frac{\Omega a}{2c_{\infty }},$$

$c_{\infty }$ being the ambient sound speed. The eigenvalue problems for the linearized Euler equations were solved numerically in [17] for various vorticity distributions, flow field calculations based on the linear theory being carried out.

Obviously, the linear theory can describe the behavior of unstable modes up to relatively small values of time $t$. So the question arise what happens with the Rankine vortex (in general, with an isolated vortex) described by the Euler equations for large values of $t$. The answer to the question is relevant to real life cases of viscous flows once the characteristic time of the vortex diffusion is considerably greater than $t$.

There are certain difficulties when attempting to numerically simulate the behavior of unstable isolated vortices during large time intervals. First of all, the amount of harmonics resolved for example by a finite difference scheme with mesh size $h$ is limited by the relation $kh<\pi$ where $k$ is the wave number. Thus refined meshes are required to meet that condition for describing the impact of high-order modes on the instability development. Unfortunately, the condition is not sufficient for getting accurate solutions for large $t$ due to accumulation with time of phase and (possibly) amplitude errors introduced by any scheme. Thus only schemes with small errors of that type can successfully accomplish it, the smaller being the errors, the greater being the admissible $t$ values.

An attempt to perform direct numerical simulations of the flow generated by the disturbed vortex with the boundary given by Eq. (1) for $n=2$ was reported in [28]. The second-order scheme [8] was applied to the Euler equations, the initial data being based on the well known exact solution for the incompressible elliptic vortex. It was found that the obtained flow field fluctuations at least for 100 rotations periods are characterized by the frequencies and the growth increments which agree well with the value from [2] and the value defined by Eq. (3) respectively. In-depth study of various isolated vortices presented in [3] contains also the results of a numerical simulation of the instability. The calculations based on the Euler equations for the local Mach number equal to 1.5 were aimed at clarifying the nonlinear stage of the flow field development. It was found that the initially disturbed by the second eigenmode vortex splits after some time moments into two corotating vortices.

Both calculations [3], [28] were limited to relatively modest time intervals (less than 1000 dimensionless time units $t$). They do not reveal the details of the modes excitations. To the best of our knowledge, the complete description of the vortex behavior and sound radiation up to considerably larger values of $t$ is lacking in the relevant literature.

The high-order multioperators-based schemes provide a good opportunity of long time direct numerical simulations in the area of computational aeroacoustic and turbulence modelings. It is due to their potential for producing very small phase and amplitude errors not only in the range of long and medium waves harmonics, but in the range of the shortest waves supported by meshes as well.

The main aim of the present study is to investigate the scenario of the unstable behavior of the subsonic Rankine vortex and its sound radiation during large time intervals using the advantages of high resolution dispersion-preserving multioperators-based schemes. The Rankine vortex was chosen because there are in-depth studies concerning its linear instability thus allowing to compare numerical and analytical data.

Both numerical method and the obtained results are presented in the following Sections.

In Section 2, the multioperators-based scheme used in the calculations is briefly outlined, the emphasize being placed on its optimization. The numerical data describing the vortex instability development and the resulting sound radiation are presented in Section 2. Finally, Section 3 the summarizes the findings and main conclusions. Appendix A Details of the multioperators, Appendix B Optimization procedures contain some details of the multioperators and the optimization procedure.