Interaction between a normal shock wave and a turbulent boundary layer at high transonic speeds. Part I: Pressure distribution

Summary Asymptotic solutions are derived for the pressure distribution in the interaction of a weak normal shock wave with a turbulent boundary layer. The undisturbed boundary layer is characterized by the law of the wall and the law of the wake for compressible flow. In the limiting case considered, for high transonic speeds, the sonic line is very close to the wall. Comparisons with experiment are shown, with corrections included for the effect of longitudinal wall curvature and for the boundary-layer displacement effect in a circular pipe.

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Zusammenfassung Asymptotische Lösungen für den Druckverlauf bei der Wechselwirkung zwischen einem schwachen normalen Stoss und einer turbulente Grenzschicht werden hergeleitet. Das Wandgesetz und Geschwindigkeitsdefekt-Gesetz für kompressible Strömung kennzeichnen die ungestörte Grenzschicht. Der Grenzfall hoher transsonischen Strömung, in dem die Schallinie in der Nähe der Wand liegt, wird untersucht. Die theoretischen Ergebnisse werden mit Experimenten verglichen. Dabei wird die Wandkrümmung und im Fall der Rohrströmung die Verdrängungsdicke berücksichtigt.

     

Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation

In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier-Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483-500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method.     

The effects of suction on the nonlinear stability of a three-dimensional compressible boundary layer

Permanent address (for correspondence), School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK. This study follows on from work by Bassom & Seddougui (1992,Proc. R. Soc. Lond. A 436, 405–15) on the effects of suctionon the nonlinear stability of the three-dimensional incompressibleboundary layer induced by a rotating disc. This flow has twotypes of stationary instability, one corresponds to the upper-branchinviscid mode and the other to the lower-branch viscous mode.This latter instability is characterized by an effective velocityprofile which has a zero shear stress at the wall and, as inBassom & Seddougui (1992), it is on this mode that interestis focused here. The effect of suction on the compressible flow,and its subsequent instability, is shown to be significant.     

Asymptotic calculation of inviscidly absolutely unstable modes of the compressible boundary layer on a rotating disk

In this work a long-wavelength asymptotic approach is used to analyze the region of absolute instability in the compressible rotating disk boundary layer flow. Theoretically determined values of branch points for the occurrence of absolute instability in the compressible flow are shown to match onto the ones which are obtained via a numerical solution of the linear inviscid compressible Rayleigh equations.     

Nonlinear elliptic boundary value problem with equivalued surface in a domain with thin layer

This paper deals with certain kinds of boundary value problems with equivalued surface of nonlinear elliptic equations on a domain with thin layer. We introduce the concept of renormalized solution to this problem. Existence and uniqueness of renormalized solutions are given, and the limit behaviour of solutions is studied in this paper.     

The evolution of travelling wave-fronts in a hyperbolic Fisher model. I. The travelling wave theory

In this paper, we consider an initial-value problem for a hyperbolicFisher equation. In particular, we consider the correspondingtravelling wave problem and establish the conditions under whichpermanent-form travelling wave solutions exist. Numerical simulationsof the initial-value problem are also presented.     

Estimation of the critical viscous sub-layer in shock wave boundary layer interaction

Zusammenfassung Wenn eine kompressible Grenzschicht (laminar oder turbulent) einer Störung unterworfen wird, sei es durch eigene Instabilität oder von aussen aufgeprägt, so macht sich der Einfluss dieser Störung sowohl stromaufwärts wie stromabwärts in der Strömungsgrenzschicht bemerkbar.Lighthill nimm in seiner mathematischen Behandlung des Mechanismus der Fortpflanzung dieser Störung stromaufwärts und stromabwärts an, dass die laminare Unterschicht der Grenzschicht genug ist, um Kompressibilitätseffekte darin vernachlässigen zu können.In vorliegender Veröffentlichung wurde die Dicke dieser kritischen laminaren Unterschicht abgeschätzt, ausgehend von ähnlichkeitsbetrachtungen und unter Verwendung vorhandener Messresultate. Es wurde festgestellt, dass das Verhältnis zwischen der kritischen, zähen Unterschicht zur totalen Grenzschtdicke (ungestört) im laminaren Fall von der Grössenordnung 110 und im turbulenten Fall etwa 1100 ist.     

The direction of the secondary wave motion in the Ekman boundary layer

The secondary wave motion in the free Ekman layer, on which a constant shear stress acts, is examined by an analytic method. The results show, that the waves move away from the center of rotation.     

Interaction between a normal shock wave and a turbulent boundary layer at high transonic speeds. Part II: Wall shear stress

Summary Asymptotic methods are used to calculate the shear stress at the wall for the interaction between a normal shock wave and a turbulent boundary layer on a flat plate. A mixing length model is used for the eddy viscosity. The shock wave is taken to be strong enough that the sonic line is deep in the boundary layer and the upstream influence is thus very small. It is shown that unlike the result found for laminar flow an asymptotic criterion for separation is not found; however, conditions for incipient separation are computed numerically using the derived solution for the shear stress at the wall. Results are compared with available experimental measurements.

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Zusammenfassung Die Schubspannung entlang einer ebenen Platte wird mit asymptotischen Methoden gerechnet für den Fall, dass ein senkrechter Stoss auf die turbulente Grenzschicht einwirkt. Für die Rechnung wird ein Mischweg-Modell benützt. Der Stoss ist von solcher Stärke, dass die Schallinie tief in der Grenzschicht liegt, und so die Wirkung stromaufwärts sehr klein ist. Im Gegensatz zu der laminaren Strömung kann man in diesem Fall eine asymptotische Bedingung für Grenzschicht-Ablösung nicht finden. Aber die Bedingungen für den Beginn der Ablösung kann man von der Lösung für die Schubspannung an der Wand numerisch erhalten. Die Ergebnisse der Theorie werden mit den vorhandenen experimentellen Messresultaten verglichen.

     

Initial value problems and first boundary problems for a class of quasilinear wave equations

The initial value problems and the first boundary problems for the quasilinear wave equation $$u_{tt} - \left[ {a_0 + na_1 \left( {u_x } \right)^{n - 1} } \right]u_{xx} - a_2 u_{xxtt} = 0$$ are considered, wherea ₀,a ₂ > 0 are constants,a ₁ is an arbitrary real number,n is a natural number. The existence and uniqueness of the classical solutions for the initial value problems and the first boundary problems of the equation (1) are proved by the Galerkin method.     

Nonlinear convection-dispersion models with a distributed pollutant source I: Direct initial boundary value problems

This paper deals with the modeling and solution of a class of nonlinear direct problems related to a transport diffusion model with a source term. Specifically, the first part of the paper deals with the derivation of a class of transport and diffusion models (with a distributed source term) in one space dimensions with variable properties along the channel and nonlinear decay term. The second part with simulations, that is the approximation to the solution of nonlinear initial boundary value problems by generalized collocation methods. The third part develops a critical analysis mainly addressed to research perspectives on the solution of inverse problems related to the identification of the source term.     

On the transition mode characterizing the triggering of a vibrator in the subsonic boundary layer on a plate

The problem of the development of two-dimensional linear perturbations in a boundary layer, generated by the triggering of a vibrator, is considered. Fourier transformations in the longitudinal coordinate and a Laplace transform in time are used to construct the solution. The inverse transforms are evaluated for large values of the characteristic time t and all values of the longitudinal coordinate x. Domains located downstream of the vibrator are studied in the first of which the perturbations will have the form of Tollmien-Schlichting waves that go over into a wave packet in the second domain. The identity in the structure of the wave packets, which are orthonormalized to the maximum amplitude for this packet for different frequencies of vibrator oscillation is noted.     

A boundary layer problem arising in gravity-driven laminar film flow of power-law fluids along vertical walls

A rigorous mathematical analysis is given for a boundary layer problem for a third-order nonlinear ordinary differential equation, which arises in gravity-driven laminar film flow of power-law fluids along vertical walls. Firstly, the problem is transformed into a singular nonlinear two-point boundary value problem of second order. Next, the latter is proved to have a unique positive solution, for which some estimates are established. Finally, the result above-mentioned is turned over to the original problem. The conclusion of this paper is that the boundary layer problem has a unique normal solution if the power-law index is less than or equal to one and a generalized normal solution if the power-law index is greater than one. Also the asymptotic behavior of the normal solution at the infinity is displayed.The work was supported by NNSF of China.     

Wave diffraction by a strip with first and second kind boundary conditions: the real wave number case

We prove unique existence of solution for a class of plane wave diffraction problems by a strip with first and second kind boundary conditions. This is done in a Bessel potential spaces framework, and for a real (noncomplex) wave number. At the end, results about the regularity (and data dependence) of the solution are exhibited upon the initial setting and the boundary parameters. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical investigation of the nonlinear wave stability of vertical thermal boundary layer flow in a porous medium

A two-dimensional, nonlinear, time-dependent, elliptic, numerical method coupled with an appropriate coordinate transformation is used to investigate the stability of free convection induced by an isothermally heated semi-infinite surface embedded in a fluid-saturated porous medium. It is found that the basic boundary layer flow is stable even to large amplitude disturbance for nondimensional distances of up to 1024 from the leading edge of the heated surface.