Numerical Modelling of Boundary Layers and Far Field Acoustic Propagation in Thermoviscous Fluid

To model linear acoustics in a thermoviscous fluid in open domain and time-harmonic regime, a Finite Element formulation in a bounded meshed domain is combined with the integral representation of the field for the propagative solution. The integrals are non-singular and involve the only Finite Element node values for temperature variation and particle velocity variables. To overcome the non-uniqueness of solutions at fictitious resonant frequencies, a Burton-Miller combination of integral representation is used. This formulation is suitable to compute acoustic radiation, scattering and diffraction by objects or mutual interaction between transducers. Two-dimensional computational experiments are presented in an infinite, open domain (exterior), showing that the model can be achieved in meshing only a thin domain surrounding the physical boundaries of a device.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Research of eigenfuctions perturbation of the transverse component velocity thermoviscous liquids flow

The viscous model fluid flow in a plane channel with a linear temperature profile is considered. The problem of the thermoviscous fluid flow stability is solved on the basis of the previously obtained generalized Orr–Sommerfeld equation by the spectral method of decomposition into Chebyshev polynomials. We study the effect of taking into account the linear and exponential dependences of the viscosity of a liquid on temperature on the eigenfunctions of the hydrodynamic stability equation and on perturbations of the transverse velocity of an incompressible fluid in a plane channel when various wall temperatures are specified. Eigenfunctions are found numerically for two eigenvalues of the linear and exponential dependence of viscosity on temperature. Presented pictures of their own functions. The eigenfunctions demonstrate the behavior of the transverse velocity perturbations, their possible growth or attenuation over time. For the given eigenfunctions, perturbations of the transverse flow velocity of a thermoviscous fluid are obtained. It is shown that taking the temperature dependence of viscosity into account affects the eigenfunctions of the equations of hydrodynamic stability and perturbations of the transverse flow velocity. Perturbations of the transverse velocity significantly affect the hydrodynamic instability of the fluid flow. The results show that when considering the unstable eigenvalue over time, the velocity perturbations begin to grow, which leads to turbulence of the flow. The maximum values of the eigenfunctions and perturbations of the transverse velocities are shifted to the hot wall. It is seen that for an unstable eigenvalue, the perturbations of the transverse flow velocity increase over time, and for a stable one, they decay.