Incorporating Linear Dynamics and Strong Anisotropy in KS. Application to Diffusion in Rotating, Stratified, MHD Turbulence, and to Aeroacoustics

Abstract

We discuss the merits of the Kinematic Simulation model for representing turbulence statistics, and especially the two-time velocity correlations in relation with Lagrangian dispersion or the acoustic emission of turbulence. We generalize the model to account for anisotropy, along two main axes. First, the discretization technique itself is modified to adapt automatically to anisotropic kinetic energy spectra, prescribed in the model. We discuss how the introduction of unsteadiness can be done from the classical assumption by relating the space and time spectra, but also by associating each wave vector with a pseudo dispersion frequency. In so doing, we show the importance of both the inclusion of a deterministic and a stochastic parts. In addition, the unsteadiness is related to the specific choice of a timescale associated with the sweeping or the straining hypothesis. A KS^new model is developed, in order to improve the original version (referred as KS^orig in the following) along two lines: first, the randomization of the wave vector is more general, and it is possible to prescribe the fully anisotropic two-point second order statistics (energy-polarization-helicity) and not only the spherically averaged energy spectrum. Second, we present how to include in the KS^new model the explicit linear dynamics associated with the distortion of turbulence by external forces. Analytical solutions provided by the Rapid Distortion Theory are used, and we show that the thus-modeled velocity fields incorporate several anisotropic features that are consistent with the physics of e.g. rotating stratified turbulence or magneto-hydrodynamics turbulence. The corresponding detailed characterization of general axisymmetric anisotropy in spectral space is of general relevance in turbulent fields produced by Direct Numerical Simulations.

Appendices

Appendix 1: General Description of Anisotropy in Fourier Space

In the following and for sake of simplicity, we drop the explicit k dependence.

Equation (24) can be derived from the decomposition of $\hat {\boldsymbol{u}}$ in terms of N and N ^∗ in (5), using $N^{*}_{i} N_{j} = P_{ij} + \mathrm{i} \varepsilon_{ijn} k_{n}/k$, with $e^{(1)}_{i} e^{(1)}_{j} + e^{(2)}_{i} e^{(2)}_{j} + \frac{k_{i} k_{j}}{k^{2}} =\delta_{ij}$ and $\mathrm{i}(e^{(1)}_{i} e^{(2)}_{j} - e^{(1)}_{j} e^{(2)}_{i}) = \mathrm{i}\varepsilon_{ijn}k_{n}/k$, because the frame of reference (e ⁽¹⁾,e ⁽²⁾,e ⁽³⁾=k/k) is orthonormal and right-handed. Accordingly, the real part of $N^{*}_{i} N_{j}$ generates the ‘energy’ part, its imaginary part does the ‘helicity’ part, and the terms N _i N _j and $N^{*}_{i} N^{*}_{j}$ generate the ‘polarization’ part. Conversely, the basic scalar spectra are derived from any arbitrarily anisotropic spectral tensor, as follows

(31)

In addition, some remarks can be made on cross-spectra related to $\hat {\mathbf{v}}^{*}(\boldsymbol{k}, t)\hat{\boldsymbol{u}}(\boldsymbol {k}, t)$, u≠v. Starting from the homogeneous average

(32)

for defining the cross-spectral tensor $F^{uv}_{ij}(\boldsymbol{k})$, this is consistent with the spatial average

(33)

From the algebraic viewpoint, the result is the same using the ‘wrong’ relationship $F^{uv}_{ij} (\boldsymbol{k}) = \langle\hat {v}^{*}_{i}(\boldsymbol{k})\hat{u}_{i}(\boldsymbol{k})\rangle$. For instance, if v is the vorticity, is found $\hat{\omega}_{i} = \mathrm{i} \varepsilon_{inj} k_{n} \hat{u}_{j}$, so that

and the helicity spectrum is recovered as

(34)

Appendix 2: Towards Cross-spectra in Stably-Stratified Flows

In addition to the previous set of spectral scalars related to double velocity correlation, it is necessary to introduce the spectrum e ^(p) of potential energy and spectral terms related to toroidal and poloidal buoyancy fluxes. For the simplest symmetry conserved by dynamical equations, or axisymmetry with mirror symmetry, Z is real (no ‘stropholysis’ spectrum in the terminology of Kassinos), is zero, and the toroidal spectral flux is zero. Given the exact conservation laws, the combination of these quantities is given by e ^(tor)=e−ℜZ, or toroidal energy spectrum, e ^w=e+ℜZ+e ^(p), or total wave energy spectrum, and an imbalance term Z′=(1/2)(e ^(tor)−e ^(pol))−iB, with B the spectrum of poloidal buoyancy flux.

Appendix 3: Towards Cross-spectra in MHD

We consider both velocity u and magnetic field b fluctuations, both solenoidal.

It is clear that the spectral tensor related to 〈b _i b _j 〉, denoted B _ij (k,t) hereinafter, has exactly the same structure as $\hat{R}_{ij}$, or

(35)

in which the quantities with superscript M are the ‘magnetic’ counterparts of the ‘kinetic’ ones (energy, polarization, helicity).

The kinetic/magnetic cross-spectra C _ij (k,t) are related to 〈b _i u _j 〉 and their structure derives from $\hat{b}^{*}_{i} \hat{u}_{j}$. From the similar decomposition of $\hat{\boldsymbol{b}}$ and $\hat{\boldsymbol {u}}$ in terms of N and N ^∗ one finds

with four pseudo-scalars, probably all complex. C _ii is the spectrum of 〈b _i u _i 〉 (cross-helicity), C ¹ is exactly half the spectrum of cross-helicity.

Another relevant term is the vector spectrum of the averaged electromotive force 〈u×b〉; it is given by −ε _mij C _ij , which reduces to $\varepsilon_{mij} \varepsilon_{ijn} \frac{k_{n}}{k} C_{2}$, so that $- 2 \mathrm{i} \frac{\boldsymbol{k}}{k} C^{2}$ is the vectorial spectrum of 〈u×b〉. Accordingly

(36)

It is unfortunately not possible to attribute to the ‘cross-polarization’ complex pseudo-scalars Z ¹ and Z ² more specific ‘physical’ meaning.

On the other hand, it is clear that all the information for second-order two-point statistics can be generated by the following list: