A. Kluwick

On the propagation of waves exhibiting both positive and negative nonlinearity

One-dimensional small-amplitude waves in which the local value of the fundamental derivative changes sign are examined. The undisturbed medium is taken to be a Navier–Stokes fluid which is at rest and uniform with a pressure and density such that the fundamental derivative is small. A weak shock theory is developed to treat inviscid motions, and the method of multiple scales is used to derive the nonlinear parabolic equation governing the evolution of weakly dissipative waves. The latter is used to compute the viscous shock structure. New phenomena of interest include shock waves having an entropy jump of the fourth order in the shock strength, shock waves having sonic conditions either upstream or downstream of the shock, and collisions between expansion and compression shocks. When the fundamental derivative of the undisturbed media is identically zero it is shown that the ultimate decay of a one-signed pulse is proportional to the negative 1/3-power of the propagation time.

Transonic nozzle flow of dense gases

The paper deals with the flow properties of dense gases in the throat area of slender nozzles. Starting from the Xavier-Stokes equations supplemented with realistic equations of state for gases which have relatively large specific heats a novel form of the viscous transonic small-perturbation equation is derived. Evaluation of the inviscid limit of this equation shows that three sonic points rather than a single sonic point may occur during isentropic expansion of such media, in contrast to the case of perfect gases. As a consequence, a shock-free transition from subsonic to supersonic speeds cannot, in general, be achieved by means of a conventional converging-diverging nozzle. Nozzles leading to shock-free flow fields must have an unusual shape consisting of two throats and an intervening antithroat. Additional new results include the computation of the internal thermoviscous structure of weak shock waves and a phenomenon referred to as impending shock splitting. Finally, the relevance of these results to the description of external transonic flows is discussed briefly.

The effect of three-dimensional obstacles on marginally separated laminar boundary layer flows

We consider the steady viscous/inviscid interaction of a two-dimensional, nearly separated, boundary layer with an isolated three-dimensional surface-mounted obstacle, for example in the large Reynolds number flow around the leading edge of a slender airfoil at a small angle of attack. An integro-differential equation describing the effect of the obstacle on the wall shear stress valid within the interaction regime is derived and solved numerically by means of a spectral method, which is outlined in detail. Typical solutions of this equation are presented for different values of the spanwise width B of the obstacle including the limiting cases B → 0 and B → ∞. Special emphasis is placed on the occurrence of non-uniqueness. On the main (upper) solution branch the disturbances to the flow field caused by the obstacle decay in the lateral direction. Conversely a periodic flow pattern, having no decay in the spanwise direction, was found to form on the lower solution branch. These branches are connected by a bifurcation point, which characterizes the maximum (critical) angle of attack for which a solution of the strictly plane interaction problem exists. An asymptotic investigation of the interaction equation, in the absence of any obstacle, for small deviations of this critical angle clearly reflects the observed behaviour of the numerical results corresponding to the different branches. As a result we can conclude that the primarily local interaction process breaks down in a non-local manner even in the limit of vanishing (three-dimensional local) disturbances of the flow field.

Dissipative waves in fluids having both positive and negative nonlinearity

The present study examines weakly dissipative, weakly nonlinear waves in which the fundamental derivative changes sign. The undisturbed state is taken to be at rest, uniform and in the vicinity of the 0 locus. The cubic Burgers equation governing these waves is solved numerically; the resultant solutions are compared and contrasted to those of the invisced theory. Further results include the presentation of a natural scaling law and inviscid solutions not reported elsewhere.

Numerical Simulation of Dry-Snow Avalanche Flow over Natural Terrain

A physical model for dry snow avalanche flow is presented. The well established model for dense granular flow proposed by Savage and Hutter [21] is adapted to describe the dense, lower part of a snow avalanche. In order to account for the high velocities of snow avalanches, a velocity dependent bed shear stress in addition to the Coulomb law for dry friction is introduced. Since the model has to be applied to arbitrarily shaped topographies, certain simplifying assumptions of geometrical nature must be introduced. In contrast to former numerical implementations of the Savage-Hutter model based on Finite Difference schemes, a Lagrangian Finite Volume method is formulated using integral balance laws. For the powder snow avalanche forming on top of the denser part a mixture model with a constant slip velocity between ice particles and air is introduced. The k-є turbulence model modified in order to account for buoyancy effects caused by the suspended particles is implemented. The resulting system of equations is solved by applying a Finite Volume scheme on a fixed grid. The transfer of snow mass from the dense flow into the powder snow avalanche is modelled by an analogy to turbulent momentum transfer.

Nonlinear waves in materials with mixed nonlinearity

Abstract The behaviour of nonlinear waves propagating in materials exhibiting mixed nonlinearity is examined. The effects of geometric spreading, diffraction and caustic formation are considered and model nonlinear evolution equations are derived. Appropriate shock conditions are constructed.

Recent Advances in Boundary Layer Theory

K. Gersten: Introduction to Boundary-Layer Theory.- H. Herwig: Laminar Boundary Layers: Asymptotic and Scaling Considerations.- R.J. Bodonyi: Boundary-Layer Stability - Asymptotic Approaches.- H. Herwig, J. Severin: The Effect of Heat Transfer on Flow Stability.- K. Gersten: Turbulent Boundary Layers I: Fundamentals.- J.D.A. Walker: Turbulent Boundary Layers II: Further Developments.- A. Kluwick: Interacting Laminar and Turbulent Boundary Layers.

Turbulent Marginal Separation and the Turbulent Goldstein Problem

A new rational theory of incompressible turbulent boundary-layer flows having a large velocity defect is presented on basis of the Reynolds-averaged Navier-Stokes equations in the limit of infinite Reynolds number. This wake-type formulation allows for, among others, the prediction of singular solutions of the boundary-layer equations under the action of a suitably controlled adverse pressure gradient, which are associated with the onset of marginally separated flows. Increasing the pressure gradient locally then transforms the marginal-separation singularity into a weak Goldstein-type singularity occurring in the slip velocity at the base of the outer wake layer. Interestingly, this behavior is seen to be closely related to (but differing in detail from) the counterpart of laminar marginal separation, in which the skin friction replaces the surface slip velocity. Most important, adopting the concept of locally interacting boundary layers results in a closure-free and uniformly valid asymptotic description of boundary layers that exhibit small, closed reverse-flow regimes. Numerical solutions of the underlying triple-deck problem are discussed.

Freely interacting transonic boundary layers

Transonic free interactions for compressive and expansive boundary‐layer flows are studied numerically and analytically. The results are qualitatively the same as those in supersonic interacting flows. However, it is found that the upstream decay is either algebraic or exponential, depending on whether the transonic interaction parameter is zero or not. The asymptoptic structure of the singularity in expansive free interactions is derived and comparison with the numerical computations shows good agreement.

Weakly nonlinear kinematic waves 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 mechanism of shock splitting and the properties of the shock structure.