The numerical computation of turbulent flows

Fundamental Principles Laminar Boundary Layer Flow Laminar Duct Flow External Natural Convection Internal Natural Convection Transition to Turbulence Turbulent Boundary Layer Flow Turbulent Duct Flow Free Turbulent Flows Convection with Change of Phase Mass Transfer Convection in Porous Media.

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Convection Heat Transfer

A theoretical framework is proposed for the analysis of adhesion between cells or of cells to surfaces when the adhesion is mediated by reversible bonds between specific molecules such as antigen and antibody, lectin and carbohydrate, or enzyme and substrate. From a knowledge of the reaction rates for reactants in solution and of their diffusion constants both in solution and on membranes, it is possible to estimate reaction rates for membrane-bound reactants. Two models are developed for predicting the rate of bond formation between cells and are compared with experiments. The force required to separate two cells is shown to be greater than the expected electrical forces between cells, and of the same order of magnitude as the forces required to pull gangliosides and perhaps some integral membrane proteins out of the cell membrane.

https://science.sciencemag.org/content/sci/200/4342/618.full.pdf

Models for the specific adhesion of cells to cells

It is shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to the streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamlines moving in the direction opposite to V. Here, a denotes the radius of the sphere, κ the magnitude of the velocity gradient, and μ and v the viscosity and kinematic viscosity, respectively. The relevance of the result to the observations by Segree & Silberberg (1962) of small spheres in Poiseuille flow is discussed briefly. Comments are also made about the problem of a sphere in a parabolic velocity profile and the functional dependence of the lift upon the parameters is obtained.

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The lift on a small sphere in a slow shear flow

A comprehensive and critical review of closure approximations for two-equation turbulence models has been made. Particular attention has focused on the scale-determining equation in an attempt to find the optimum choice of dependent variable and closure approximations. Using a combination of singular perturbation methods and numerical computations, this paper demonstrates that: 1) conventional A:-e and A>w formulations generally are inaccurate for boundary layers in adverse pressure gradient; 2) using "wall functions'' tends to mask the shortcomings of such models; and 3) a more suitable choice of dependent variables exists that is much more accurate for adverse pressure gradient. Based on the analysis, a two-equation turbulence model is postulated that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows. With no viscous damping of the model's closure coefficients and without the aid of wall functions, the model equations can be integrated through the viscous sublayer. Surface boundary conditions are presented that permit accurate predictions for flow over rough surfaces and for flows with surface mass addition.

Reassessment of the scale-determining equation for advanced turbulence models

The linear stability of the symmetric Burgers vortex to a class of two-dimensional perturbations is demonstrated The linear stability of the axially symmetric Burgers vortex to two-dimensional low-Reynolds number disturbances is first discussed The analysis describes the connection of the normal modes with a classical set of special functions An expansion in Reynolds number gives the variation of the temporal eigenvalues of the normal modes Then, Burgers-type solutions are sought where the radially inward-straining field is not axially symmetric A rational perturbation procedure is used to obtain a new class of nonsymmetric Burgers vortices valid when the Reynolds number and the asymmetry in the external strain are small Finally, the perturbation solutions are numerically extended to larger values of the parameters

Stability and structure of stretched vortices

The structure of the intense vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers in the range Re(sub lambda) = 36-171. In accordance with previous investigators, this vorticity is found to be organized in coherent, cylindrical or ribbon-like, vortices ('worms'). A statistical study suggests that they are just especially intense features of the background, O(omega'), vorticity. Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow. An interesting observation is that the Reynolds number based on the circulation of the intense vortices, gamma/nu, increases monotonically with Re(sub lambda), raising the question of the stability of the structures in the limit of Re(sub lambda) approaching infinity. One and two-dimensional statistics of vorticity and strain are presented; they are non-gaussian, and the behavior of their tails depends strongly on the Reynolds number. There is no evidence of convergence to a limiting distribution in our range of Re(sub lambda), even though the energy spectra and the energy dissipation rate show good asymptotic properties in the higher Reynolds number cases. Evidence is presented to show that worms are natural features of the flow and that they do not depend on the particular forcing scheme.

The structure of intense vorticity in homogeneous isotropic turbulence

The stability of the finite-area Karman ‘vortex street’ to two-dimensional disturbances is determined. It is shown that for vortices of finite size there exists a finite range of spacing ratio κ for which the array is stable to infinitesimal disturbances. As the vortex size approaches zero, the range narrows to zero width about the classical von Karman value of 0·281.

/pdf/stability-of-a-vortex-street-of-finite-vortices-3rvkybrygr.pdf

Stability of a vortex street of finite vortices

The steady axisymmetric Euler flow of an inviscid incompressible swirling fluid
is described exactly by the Squire-Long equation. This equation is studied numerically
for the case of diverging flow to investigate the dependence of solutions
on upstream, or inlet, and downstream, or outlet, boundary conditions and flow
geometry. The work is performed with a view to understanding how the phenomenon
of vortex breakdown occurs. It is shown that solutions fail to exist or,
alternatively, that the axial flow ceases to be unidirectional, so that breakdown
can be inferred, when a parameter measuring the relative magnitude of rotation
and axial flow (the Squire number) exceeds critical values depending upon the geometry
and inlet profiles. A 'quasi-cylindrical' simplification of the Squire-Long
equation is compared with the more complete Euler model and shown to be able
to account for most of the latter's behaviour. The relationship is examined between
'failure' of the quasi-cylindrical model and the occurrence of a 'critical'
flow state in which disturbances can stand in the flow.

Inviscid swirling flows and vortex breakdown

The three-dimensional linear stability of a rectilinear vortex of elliptical cross-section existing as a steady state in an irrotational straining field is studied numerically in the case of finite strain. It is shown that the instability predicted analytically for weak strain persists for finite strain and that the weak-strain results continue to be quantitatively valid for finite strain. The dependence of the growth rates of the unstable modes on the strain and the axial-disturbance wavelength is discussed. It is also shown that a three-dimensional instability is always more unstable than a two-dimensional instability in the range of parameters of most interest.

/pdf/three-dimensional-stability-of-an-elliptical-vortex-in-a-q1aozha41s.pdf

Philip Geoffrey Saffman

Papers

Stability and structure of stretched vortices

The structure of intense vorticity in homogeneous isotropic turbulence

Stability of a vortex street of finite vortices

Inviscid swirling flows and vortex breakdown

Three-dimensional stability of an elliptical vortex in a straining field