The hierarchies of equations for a general multi-point probability density function (PDF) and its characteristic function (CF) are derived for compressible turbulent flows, obeying the ideal gas law. The closure problem of turbulence is clearly exhibited in each of the approaches, with n-point statistics being dependent on the (n + 1)-point statistics and, for some cases, even the (n + 2)-point statistics. When dynamic viscosity and heat conductivity are dependent on temperature as a power-law, the CF hierarchy could contain fractional derivatives if the exponent is a non-integer. The additional conditions satisfied by all the PDFs and CFs in both the hierarchies are also prescribed. The PDF and CF equations derived in this paper, with the unclosed terms explicitly written in terms of higher order PDF/CF, act as a starting point in constructing symmetry-based invariant solutions of compressible turbulence, analogous to the works of Wacławczyk et al. [“Statistical symmetries of the Lundgren–Monin–Novikov hierarchy,” Phys. Rev. E 90, 013022 (2014)] and Oberlack and Rosteck [“New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws,” Discrete Contin. Dyn. Syst. 3, 451–471 (2010)] for incompressible turbulence.
Presently, the oscillation of a liquid droplet in a dynamically negligible outer medium subject to surface tension and small viscosity is investigated. By using the potential flow assumption, the unified transform method by Fokas is employed to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations (IDEs). This new system depends on one less spatial variable and is now defined on a time-independent domain. Most importantly, the resulting set of equations governs the general droplet oscillation with arbitrarily large deviations from the spherical shape. As the nonlinearity of the above IDE system up to now prevented an analytical solution, the Poincaré expansion technique is employed, retaining terms up to the second order. By decomposing the unknowns into normal modes, these equations are uncoupled and the resulting ordinary differential equations for the mode amplitudes are solved, and the results are compared to those of previous works. It should be stressed that the present analysis is limited to small viscosity, or, in other words, for small Ohnesorge numbers. The reason for this is that, inside of the droplet, a potential flow is assumed and the viscous effect is taken into account only at the droplet surface by the jump condition of momentum. This is only reasonable for a small viscosity and a short time. Otherwise, vorticity is generated at the interface and diffuses toward the inside of the droplet.
In this paper the one-dimensional two-phase Stefan problem is studied analytically leading to a system of non-linear Volterra-integral-equations describing the heat distribution in each phase. For this the unified transform method has been employed which provides a method via a global relation, by which these problems can be solved using integral representations. To do this, the underlying partial differential equation is rewritten into a certain divergence form, which enables to treat the boundary values as part of the integrals. Classical analytical methods fail in the case of the Stefan problem due to the moving interface. From the resulting non-linear integro-differential equations the one for the position of the phase change can be solved in a first step. This is done numerically using a fix-point iteration and spline interpolation. Once obtained, the temperature distribution in both phases is generated from their integral representation.
We investigate the two-dimensional (2D) stability of rotational shear flows in an unbounded domain. The eigenvalue problem is formulated by using a novel algebraic mode decomposition distinct from the normal modes with temporal evolution $\exp(\omega t)$. Based on the work of \citeasnoun{NoldOberlack2013}, we show how these new modes can be constructed from the symmetries of the linearized stability equation. For the azimuthal base flow velocity $V(r)=r^{-1}$ an additional symmetry exists, such that a mode with algebraic temporal evolution $t^s$ is found. $s$ refers to an eigenvalue for the algebraic growth or decay of the kinetic energy of the perturbations. An eigenvalue problem for the viscous and inviscid stability using algebraic modes is formulated on an infinite domain with $r \to \infty$. An asymptotic analysis of the eigenfunctions shows that the flow is linearly stable under 2D perturbations. We find stable modes with the algebraic mode ansatz, which can not be obtained by a normal mode analysis. The stability results are in line with Rayleigh's inflection point theorem.
We compute the Lie symmetries of characteristic function (CF) hierarchy of compressible turbulence, ignoring the effects of viscosity and heat conductivity. In the probability density function (PDF) hierarchy, a typical non-local nature is observed, which is naturally eliminated in the CF hierarchy. We observe that the CF hierarchy retains all the symmetries satisfied by compressible Euler equations. Broadly speaking, four types of symmetries can be discerned in the CF hierarchy: (i) symmetries corresponding to coordinate system invariance, (ii) scaling/dilation groups, (iii) projective groups and (iv) statistical symmetries, where the latter define measures of intermittency and non-gaussianity. As the multi-point CFs need to satisfy additional constraints such as the reduction condition, the projective symmetries are only valid for monatomic gases, that is, the specific heat ratio, $\gamma = 5/3$ . The linearity of the CF hierarchy results in the statistical symmetries due to the superposition principle. For all of the symmetries, the global transformations of the CF and various key compressible statistics are also presented.
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