The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Coherent structures in wall turbulence transport momentum and provide a means of producing turbulent kinetic energy. Above the viscous wall layer, the hairpin vortex paradigm of Theodorsen coupled with the quasistreamwise vortex paradigm have gained considerable support from multidimensional visualization using particle image velocimetry and direct numerical simulation experiments. Hairpins can autogenerate to form packets that populate a significant fraction of the boundary layer, even at very high Reynolds numbers. The dynamics of packet formation and the ramifications of organization of coherent structures (hairpins or packets) into larger-scale structures are discussed. Evidence for a large-scale mechanism in the outer layer suggests that further organization of packets may occur on scales equal to and larger than the boundary layer thickness.

Hairpin vortex organization in wall turbulencea)

[1] This paper presents a new, generalized two-phase debris flow model that includes many essential physical phenomena. The model employs the Mohr-Coulomb plasticity for the solid stress, and the fluid stress is modeled as a solid-volume-fraction-gradient-enhanced non-Newtonian viscous stress. The generalized interfacial momentum transfer includes viscous drag, buoyancy, and virtual mass. A new, generalized drag force is proposed that covers both solid-like and fluid-like contributions, and can be applied to drag ranging from linear to quadratic. Strong coupling between the solid- and the fluid-momentum transfer leads to simultaneous deformation, mixing, and separation of the phases. Inclusion of the non-Newtonian viscous stresses is important in several aspects. The evolution, advection, and diffusion of the solid-volume fraction plays an important role. The model, which includes three innovative, fundamentally new, and dominant physical aspects (enhanced viscous stress, virtual mass, generalized drag) constitutes the most generalized two-phase flow model to date, and can reproduce results from most previous simple models that consider single- and two-phase avalanches and debris flows as special cases. Numerical results indicate that the model can adequately describe the complex dynamics of subaerial two-phase debris flows, particle-laden and dispersive flows, sediment transport, and submarine debris flows and associated phenomena.

/pdf/a-general-two-phase-debris-flow-model-3sijccpo55.pdf

A general two-phase debris flow model

Hydrodynamic and hydromagnetic stability

/pdf/dispersive-shock-waves-and-modulation-theory-1zjx37cug6.pdf

Dispersive shock waves and modulation theory

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.

On the propagation of waves exhibiting both positive and negative nonlinearity

The paper deals with the flow properties of dense gases in the throat area of slender nozzles. Starting from the Navier–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.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112093000618

Transonic nozzle flow of dense gases

The study of materials which exhibit new and unconventional properties is of central importance for the devel- opment of advanced and refined technologies in many fields of engineering science. In this connection there has been a rapidly growing interest in real fluid effects on wave phenomena in the past few years. A prominent example is provided by Bethe-Zel'dovich-Thompson (BZT) fluids which have the distinguishing feature that they exhibit negative nonlinearity over a finite range of temperature and pressures in the pure vapour phase. However, two phase flows with and without phase change are an even richer source of new unexpected and previously thought impossible phenomena. Topics covered by this volume include waves in gases near the critical point, waves in retrograde fluids, temperature waves in superfluid helium and density waves in suspensions of particles in liquids. Clearly, the aim of the various contributions is twofold. First, they are intended to provide scientists and engineers working in these and related areas with an overview of various new physical phenomena as for example expansion shocks, sonic shocks, shock splitting, evaporation and liquafaction shocks and the experimental techniques needed to study these phenomena. Second, an attempt is made to discuss aspects of their mathematical modeling with special emphasis on properties which these phenomena have in common.

Nonlinear waves in real fluids

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.

Numerical Simulation of Dry-Snow Avalanche Flow over Natural Terrain

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.

Alfred Kluwick

Papers

On the propagation of waves exhibiting both positive and negative nonlinearity

Transonic nozzle flow of dense gases

Nonlinear waves in real fluids

Numerical Simulation of Dry-Snow Avalanche Flow over Natural Terrain

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