Stefan Scheichl

Non-classical kinematic shocks in suspensions of particles in fluids

SummaryThe properties of weakly nonlinear kinematic waves in suspensions are investigated under the assumption that the particle concentration deviates only slightly from the value at the inflexion point of the drift flux curve. Special emphasis is placed on the conditions for the existence of an internal dissipative-dispersive shock structure. The resulting shock admissibility criteria are found to be significantly different from those following from standard theories of kinematic waves. Most interesting, the analysis shows that non-classical shocks which emanate rather than absorb characteristics may be admissible under certain conditions.

Near-critical hydraulic flows in two-layer fluids

This paper deals with the propagation of nearly resonant gravity waves in two-layer flows over a bottom topography assuming that both fluids are incompressible and inviscid. Evolution equations are derived for weakly nonlinear surface-layer and internal-layer waves in the hydraulic limit of infinite wavelength. Special emphasis is placed on the flow regime where the quadratic nonlinear parameter associated with internal-layer waves is small or vanishes. For example, this is the case for all possible density ratios if the velocities in both layers are equal and if the interface height is close to one-half the total fluid-layer height. The waves then exhibit so-called mixed nonlinearity leading in turn to the formation of positive and negative hydraulic jumps. Considerations based on a model equation for the internal dissipative-dispersive structure of hydraulic jumps indicate that the admissibility of discontinuities in this regime depends strongly on the relative magnitudes of dispersion and dissipation. Surprisingly, these admissible hydraulic jumps may violate the wave-speed-ordering relationship which requires that the upstream wave speed does not exceed the propagation speed of the discontinuity. An important example is provided by the inviscid hydraulic jump, which has been known for some time, although its non-classical nature, in that it transmits rather than absorbs waves, has apparently not been recognized before.

On the calculation of the transmission line parameters for long tubes using the method of multiple scales

The present paper deals with the classical problem of linear sound propagation in tubes with isothermal walls. The perturbation technique of the method of multiple scales in combination with matched asymptotic expansions is applied to derive the first-order solutions and, in addition, the second-order solutions representing the correction due to boundary layer attenuation. The propagation length is assumed to be so large that in order to obtain asymptotic solutions which extend over the whole spatial range the first-order corrections to the classical attenuation rates of the different modes come into play as well. Starting with the case of the characteristic wavelength being large compared to the characteristic dimension of the duct, the analysis is then extended to the case where both of these quantities are of the same order of magnitude. Furthermore, the transmission line parameters and the transfer functions relating the sound pressures at the ends of the duct to the axial velocities are calculated.

Linear and nonlinear propagation of higher order modes in hard-walled circular ducts containing a real gas

This paper deals with the linear and nonlinear propagation of sound waves through a real gas contained in a circular tube with rigid, isothermal walls. Special emphasis is placed on the asymptotically correct treatment of the higher order modes and their interaction with the acoustic boundary layer. In the first part, a linear perturbation analysis is carried out to calculate the correction terms arising from the viscothermal damping mechanisms present in the system. In extension to previous work, the propagation length is assumed to be so large that the exponentially growing boundary layer effects do not only affect the second order terms of the sound pressure but also the leading order terms. The series expansions derived for the propagation parameters extend the results given in the literature with additional terms resulting from viscosity and heat conduction in the core region. The second part is concerned with the nonlinear modulation of a wave packet transmitted through a real gas. A damped nonlinear Schrödinger equation is derived and its solutions for positive as well as negative values of the nonlinearity parameter are studied. In particular, the case of wave propagation in ducts containing a so-called BZT fluid is discussed.

On recent developments in marginal separation theory

Thin aerofoils are prone to localized flow separation at their leading edge if subjected to moderate angles of attack α. Although ‘laminar separation bubbles’ at first do not significantly alter the aerofoil performance, they tend to ‘burst’ if α is increased further or if perturbations acting upon the flow reach a certain intensity. This then either leads to global flow separation (stall) or triggers the laminar–turbulent transition process within the boundary layer flow. This paper addresses the asymptotic analysis of the early stages of the latter phenomenon in the limit as the characteristic Reynolds number , commonly referred to as marginal separation theory. A new approach based on the adjoint operator method is presented that enables the fundamental similarity laws of marginal separation theory to be derived and the analysis to be extended to higher order. Special emphasis is placed on the breakdown of the flow description, i.e. the formation of finite-time singularities (a manifestation of the bursting process), and on its resolution being based on asymptotic arguments. The passage to the subsequent triple-deck stage is described in detail, which is a prerequisite for carrying out a future numerical treatment of this stage in a proper way. Moreover, a composite asymptotic model is developed in order for the inherent ill-posedness of the Cauchy problems associated with the current flow description to be resolved.

Asymptotic analysis of surface waves in continuous strip casting processes

This paper presents a two-dimensional analysis of surface waves possibly emerging in a specific open channel flow with continuous solidification, i.e. the fluid consisting of molten material is cooled from below and solidifies. In modern metallurgical engineering such processes are of importance for the strip casting of steel and other metals. The study is based on the assumption that the wavelengths are large compared to the characteristic depth of the melt but small compared to the solidification length. Within the framework of a weakly nonlinear theory the use of the Euler equations supplemented with the appropriate boundary conditions at the solidification front and the free surface yields two Korteweg–de Vries equations with varying coefficients, which govern the propagation of the waves. However, the adopted form of the asymptotic expansions ceases to be valid as the point of complete solidification is approached, where the displacements at the free boundary and the depth of the melt are of the same order. Thus, a separate investigation for this region is carried out in order to describe the further evolution of the surface waves and its influence on the final shape of the fully solidified metal sheet.

On blow-up solutions in marginally separated triple-deck flows

The present paper deals with blow-up solutions associated with the high Reynolds number asymptotic theory of unsteady, planar and marginally separated boundary layer flows. In particular, the case is treated in which a rapid focusing process results in the breakdown of the fully nonlinear triple-deck structure that governs the next stage on from classical marginal separation. The terminal form of these equations in the event of such a blow-up were successfully solved numerically by the use of a scheme based on Chebyshev polynomials for both spatial directions. Surprisingly, the computations showed that the terminal solutions for the flow quantities are not unique, but form a two-parametric family.