Stanley A. Berger

Numerical Analysis of Flow Through a Severely Stenotic Carotid Artery Bifurcation

The results of computational simulations may supplement MR and other in vivo diagnostic techniques to provide an accurate picture of the hemodynamics in particular vessels, which may help demonstrate the risks of embolism or plaque rupture posed by particular plaque deposits. In this study, a model based on an endarterectomy specimen of the plaque in a carotid bifurcation was examined. The flow conditions include steady flow at Reynolds numbers of 300, 600, and 900 as well as unsteady pulsatile flow. Both dynamic pressure and wall shear stress are very high, with shear values up to 70 N/m2, proximal to the stenosis throat in the internal carotid artery, and both vary significantly through the flow cycle. The wall shear stress gradient is also strong along the throat. Vortex shedding is observed downstream of the most severe occlusion. Two turbulence models, the Chien and Goldberg varieties of k-epsilon, are tested and evaluated for their relevance in this geometry. The Chien model better captures phenomena such as vortex shedding. The flow distal to stenosis is likely transitional, so a model that captures both laminar and turbulent behavior is needed.

Solutions of the Navier-Stokes equations for vortex breakdown

Steady solutions of the Navier-Stokes equations, in terms of velocity and pressure, for breakdown in an unconfined viscous vortex are obtained numerically using the artificial compressibility technique of Chorin combined with an ADI finite-difference scheme. Axisymmetry is assumed and boundary conditions are carefully applied at the boundaries of a large finite region in an axial plane while resolution near the axis is maintained by a coordinate transformation. The solutions, which are obtained for Reynolds numbers up to 200 based on the free-stream axial velocity and a characteristic core radius, show that breakdown results from the diffusion and convection of vorticity away from the vortex core which, because of the strong coupling between the circumferential and axial velocity fields in strongly swirling flows, can lead to stagnation and reversal of the axial flow near the axis.

Entry flow in a curved pipe

A secondary flow is set up when a fluid flows through a stationary curved pipe. The fluid in the middle of the pipe moves outwards and that near the wall inwards. Dean showed that the dynamical similarity of this fully developed flow depends on a non-dimensional parameter

Laminar entrance flow in a curved pipe

D = 2(a/R)^{\frac{1}{2}}(a\overline{W}/\nu)

Analysis and modelling of turbulent flow in an axially rotating pipe

, where

Effects of internal heat transfer on the structure of self-similar blast waves

\overline{W}

Flow in heated curved pipes

is the mean velocity along the pipe, v is the coefficient of kinematic viscosity and a is the radius of the pipe, which is bent into a coil of radius R. Deans analysis was limited to small values of D. Later, Barua developed an asymptotic boundary-layer theory for large values of D and showed for these values of D that the resistance coefficient γ c is much larger than that for the corresponding straight pipe. The present work deals with the flow in a curved pipe as it develops from a uniformly distributed velocity at the entrance to a fully developed profile. Baruas results for the fully developed flow are adopted as downstream conditions in the present work. The ratio of the entry lengths of the curved ipe and the corresponding straight one is shown to be proportional to D −1/2 when D is large. Thus, the entry length for a curved pipe is much shorter than that for the corresponding straight pipe.

Numerical simulation of magnetic resonance angiographies of an anatomically realistic stenotic carotid bifurcation.

The full elliptic Navier–Stokes equations have been solved for entrance flow into a curved pipe using the artificial compressibility technique developed by Chorin (1967). The problem is formulated for arbitrary values of the curvature ratio and the Dean number. Calculations are carried out for two curvature ratios, a/R = 1/7 and 1/20, and for Dean number ranging from 108.2 to 680.3, in a computational mesh extending from the inlet immediately adjacent to the reservoir to the fully developed downstream region. Secondary flow separation near the inner wall is observed in the developing region of the curved pipe. The separation and the magnitude of the secondary flow are found to be greatly influenced by the curvature ratio. As observed in the experiments of Agrawal, Talbot & Gong (1978) we find: (i) two-step plateau-like axial-velocity profiles for high Dean number, due to the secondary flow separation, and (ii) doubly peaked axial-velocity profiles along the lines parallel to the plane of symmetry, due to the highly distorted secondary-flow vortex structure.

The interaction between a counter‐rotating vortex pair in vertical ascent and a free surface

The analysis and modelling of the structure of turbulent flow in a circular pipe subjected to an axial rotation is presented. Particular attention is paid to determining the terms in various turbulence closures that generate the two main physical features that characterize this flow: a rotationally dependent axial mean velocity and a rotationally dependent mean azimuthal or swirl velocity relative to the rotating pipe. It is shown that the rst feature is well represented by two-dimensional explicit algebraic stress models but is irreproducible by traditional two-equation models. On the other hand, three-dimensional frame-dependent models are needed to predict the presence of a mean swirl velocity. The latter is argued to be a secondary eect which arises from a cubic nonlinearity in standard algebraic models with conventional nearwall treatments. Second-order closures are shown to give a more complete description of this flow and can describe both of these features fairly well. In this regard, quadratic pressure{strain models perform the best overall when extensive comparisons are made with the results of physical and numerical experiments. The physical signicance of this problem and the implications for future research in turbulence are discussed in detail.

Boundary-layer theory for blast waves

Profiles of gasdynamic parameters in self-similar blast waves, taking into account the influence of conduction and radiation fluxes due to high temperatures attained at the centre, are determined. In the blast-wave equations these fluxes are expressed in terms of the Fourier law for heat conduction and a differential expression for radiative transport in a semi-grey gas model. Various boundary conditions are considered in order to account for different ways in which blast waves are initiated and driven. Similarity requirements are implemented in the solution by compatible functional forms of gas conductivity and absorptivity, as well as the opacity of the shock front. This formulation yields a two-point boundary-value problem, which is then transformed into an initial-value problem in order to facilitate the integration. As a particular example, a detailed solution for the constant-energy case is obtained, covering the whole range of relative heat-transfer effects expressed in terms of radiative to gasdynamic energy fluxes, from the adiabatic flow field, on one extreme, to the isothermal, on the other.