Viscous Mean Flow Research Articles

Linear lattice Boltzmann flux solver for simulating acoustic propagation

A linear lattice Boltzmann flux solver (LLBFS) is developed based on the high-precision finite-volume method (FVM) and is used to simulate problems involving acoustic propagation. The least-squares-based finite difference (LSFD) is used to construct the high-precision FVM, which discretizes the linear Navier–Stokes equations (LNSE). In terms of the flux calculation for the interface of the control cell, the LLBFS is derived by comparing the macroscopic and microscopic variables through the Chapman–Enskog (C-E) multi-scale analysis, which establishes the relationship between the linear lattice Boltzmann equation (LBE) and the LNSE. The algorithm has the following advantages. First, the proposed LLBFS inherits the advantages of the lattice Boltzmann flux solver (LBFS), i.e., the local solution of the lattice Boltzmann equation is used to calculate the fluxes; handling the inviscid and viscous fluxes simultaneously makes for a more consistent algorithm and it can be expanded to the multi-dimensional scheme without approximation. Second, solving the LNSE using the high-precision FVM on unstructured grids, which enables problems involving acoustic waves interacting with complex geometries can be simulated conveniently. To validate the accuracy and robustness of the proposed method, several cases are simulated, including acoustic waves propagating in an inviscid and viscous mean flow and interacting with complex geometries.

Influence of non-uniform mean flow on acoustic scattering from complex geometries

An immersed boundary computational model is presented to study the influence of non-uniform mean flow on acoustic scattering from complex geometries. It is firstly demonstrated from two benchmark problems that an absorbing treatment for the interior of solid domain in the developed immersed boundary method is not necessary. Then, simulations for acoustic scattering both from a circular cylinder and multiple bodies with different mean flow models are performed. It is found that the existence of a core wake region in a background mean flow model plays a significant role in accurately predicting acoustic directivity. When a sound source is in the core wake region, the background mean flow model has an obvious influence on the predicted acoustic directivity whether for a simple circular cylinder or multiple bodies; when the sound source is away from the core wake region, consistent results for different background mean flow models in most directions are obtained. However, about a maximum 8 dB discrepancy in the “front arm” direction of a circular cylinder and about a maximum 4.8 dB discrepancy in the downstream direction for the multi-element airfoil are caused respectively for the parallel and viscous mean flow models.

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

The mean gaseous motion in solid rocket motors has been traditionally described using an inviscid solution in a porous tube of fixed radius and uniform wall injection. This model, usually referred to as the Taylor–Culick profile, consists of a rotational solution that captures the bulk gaseous motion in a frictionless rocket chamber. In practice, however, the port radius increases as the propellant burns, thus leading to time-dependent effects on the mean flow. This work considers the related problem in the context of viscous motion in a porous tube and allows the radius to be time dependent. By implementing a similarity transformation in space and time, the incompressible Navier–Stokes equations are first reduced to a nonlinear fourth-order ordinary differential equation with four boundary conditions. This equation is then solved both numerically and asymptotically, using the injection Reynolds number and the dimensionless wall regression ratio as primary and secondary perturbation parameters. In this manner, closed-form analytical solutions are obtained for both large and small Reynolds number with small-to-moderate . The resulting approximations are then compared with the numerical solution obtained for an equivalent third-order ordinary differential equation in which both shooting and the irregular limit that affects the fourth-order formulation are circumvented. This code is found to be capable of producing the stable solutions for this problem over a wide range of Reynolds numbers and wall regression ratios.

Acoustics in a Two-Deck Viscothermal Boundary Layer over an Impedance Surface

The acoustics of a mean flow boundary layer over an impedance surface or acoustic lining is considered. By considering a thick mean flow boundary layer (possibly due to turbulence), the boundary-layer structure is separated asymptotically into two decks, with a thin weakly viscous mean flow boundary layer and an even thinner strongly viscous acoustic sublayer, without requiring a high frequency. Using this, analytic solutions are found for the acoustic modes in a cylindrical lined duct. The mode shapes in each region compare well with numerical solutions of the linearized compressible Navier–Stokes equations, as does a uniform composite asymptotic solution. A closed-form effective impedance boundary condition is derived, which can be applied to acoustics in inviscid slipping flow to account for both shear and viscosity in the boundary layer. The importance of the boundary layer is demonstrated in the frequency domain, and the new boundary condition is found to correctly predict the attenuation of upstream-propagating cut-on modes, which are poorly predicted by existing inviscid boundary conditions. Stability is also investigated, and the new boundary condition is found to yield good results away from the critical layer. A time-domain formulation of a simplified version of the new impedance boundary condition is proposed.

How rapidly oscillating collapsible tubes extract energy from a viscous mean flow

We present a combined theoretical and computational analysis of three-dimensional unsteady finite-Reynolds-number flows in collapsible tubes whose walls perform prescribed high-frequency oscillations which resemble those typically observed in experiments with a Starling resistor. Following an analysis of the flow fields, we investigate the system's overall energy budget and establish the critical Reynolds number,Recrit, at which the wall begins to extract energy from the flow. We conjecture thatRecritcorresponds to the Reynolds number beyond which collapsible tubes are capable of performing sustained self-excited oscillations. Our computations suggest a simple functional relationship betweenRecritand the system parameters, and we present a scaling argument to explain this observation. Finally, we demonstrate that, within the framework of the instability mechanism analysed here, self-excited oscillations of collapsible tubes are much more likely to develop from steady-state configurations in which the tube is buckled non-axisymmetrically, rather than from axisymmetric steady states, which is in agreement with experimental observations.

Modulated surface waves in large-aspect-ratio horizontally vibrated containers

We consider the harmonic and subharmonic modulated surface waves that appear upon horizontal vibration along the surface of the liquid in a two-dimensional large-aspect-ratio (length large compared to depth) container, whose depth is large compared to the wavelength of the surface waves. The analysis requires us also to consider an oscillatory bulk flow and a viscous mean flow. A weakly nonlinear description of the harmonic waves is made which provides the threshold forcing amplitude to trigger harmonic instabilities, which are of various qualitatively different kinds. A linear analysis provides the threshold amplitude for the appearance of subharmonic waves through a subharmonic instability. The results obtained are used to make several specific qualitative and quantitative predictions.

Viscous Faraday waves in two-dimensional large-aspect-ratio containers

A weakly nonlinear analysis of one-dimensional viscous Faraday waves in two-dimensional large-aspect-ratio containers is presented. The surface wave is coupled to a viscous long-wave mean flow that is slaved to the free-surface deformation. The relevant Ginzburg–Landau-like amplitude equations are derived from first principles, and can be of three different types, depending on the ratio between wavelength, depth and the viscous length. These three equations are new in the context of Faraday waves. The coefficients of these equations are calculated for arbitrary viscosity and compared with their counterparts in the literature for small viscosity; a discrepancy in the cubic coefficient is due to a dramatic sensitivity of this coefficient on a small wavenumber shift due to interplay between viscous effects and parametric forcing.

Standing wave description of nearly conservative, parametrically driven waves in extended systems

We consider the standing wavetrains that appear near threshold in a nearly conservative, parametrically excited, extended system that is invariant under space translations and reflection. Sufficiently close to threshold, the relevant equation is a Ginzburg–Landau equation whose cubic coefficient is extremely sensitive to wavenumber shifts, which can only be understood in the context of a more general quintic equation that also includes two cubic terms involving the spatial derivative. This latter equation is derived from the standard system of amplitude equations for counterpropagating waves, whose validity is well established today. The coefficients of the amplitude equation for standing waves are obtained for 1D Faraday waves in a deep container, to correct several gaps in former analyses in the literature. This application requires to also consider the effect of the viscous mean flow produced by the surface waves, which couples the dynamics of the surface waves themselves with the free surface deformation induced by the mean flow.

Turbulent thermal convection in a closed domain: viscous boundary layer and mean flow effects

In this paper the effects of viscous boundary layers and mean flow structures on the heat transfer of a flow in a slender cylindrical cell are analysed using the direct numerical simulation of the Navier-Stokes equations with the Boussinesq approximation. Ideal flows are produced by suppressing the viscous boundary layers and by artificially enforcing the flow axisymmetry with the aim of checking some proposed explanations for the Nusselt number dependence on the Rayleigh number. The emerging picture suggests that, in this slender geometry,the presence of the viscous boundary layers does not have appreciable impact on the slope of the Nu vs. Ra relation while a transition of the mean flow is most likely the reason for the slope increase observed around Ra=2 x 109.

Weakly dissipative Faraday waves in 2D large aspect ratio annuli

Weakly dissipative parametrically excited (by vertical vibration) surface gravity-capillary waves in a two-dimensional, horizontally periodic container are considered. A set of equations is derived for the coupled evolution of the left- and right-traveling surface waves and the associated mean flow, in the case when the container depth is small compared to its length but large compared to the wavelength of the excited waves. The stability of the spatially uniform standing waves (SWs) is first analyzed and then the large time spatio-temporal behavior of the system beyond threshold is numerically studied. The viscous mean flow is found to drastically affect the dynamics of the system and the resulting surface wave patterns.

Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio

Nearly inviscid parametrically excited surface gravity–capillary waves in two-dimensional domains of finite depth and large aspect ratio are considered. Coupled equations describing the evolution of the amplitudes of resonant left- and right-traveling waves and their interaction with a mean flow in the bulk are derived, and the conditions for their validity established. Under suitable conditions the mean flow consists of an inviscid part together with a viscous mean flow driven by a tangential stress due to an oscillatory viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer. These forcing mechanisms are important even in the limit of vanishing viscosity, and provide boundary conditions for the Navier–Stokes equation satisfied by the mean flow in the bulk. For moderately large aspect ratio domains the amplitude equations are nonlocal but decouple from the equations describing the interaction of the slow spatial phase and the viscous mean flow. Two cases are considered in detail, gravity–capillary waves and capillary waves in a microgravity environment.

An analytical model for ovalling oscillation of clamped-clamped cylindrical shells in cross flow

This paper presents a quasi-static aeroelastic theory for predicting the ovalling oscillations of thin cylindrical shells in cross flow. The flow is treated as a superposition of the measured viscous mean flow and a potential flow associated with deformation of the shell. The existence of the wake is taken into account, but flow within it associated with the formation of vortices, is entirely ignored. The aerodynamic forces are formulated by means of strip theory and the dynamics of the shell are described by means of Flugge's equations. It is found that at sufficiently high flow velocities, the shell becomes subject to oscillatory instabilities associated with negative aerodynamic damping. The principal observed characteristics of ovalling oscillation are qualitatively predicted by theory, but critical flow velocities are overestimated by a factor ranging from 1·6 to 5.

Propagation of acoustic waves in ducts with varying cross sections and viscous mean flow

Abstract A numerical analysis is presented of the propagation of acoustic waves in a two-dimensional duct with sinusoidally varying walls. In the generalized case of a duct carrying viscous mean flow at low Mach numbers the results show that resonance occurs for specific values of the basic parameters, such as the wavenumber of the wall, the frequency of sound and the Mach number of the mean flow. The numerical model, by which any form of continuously varying walls can be treated, includes a finite difference method and a Newton—Raphson technique to obtain the resonant conditions.