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.

/pdf/the-lift-on-a-small-sphere-in-a-slow-shear-flow-1ltpy7jdp5.pdf

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

Calculations of a discrete nonlinear dispersive wave system show that as the degree of nonlinearity increases, the system experiences in turn, periodic, recurring, chaotic, transitional, and periodic motions. A relationship between the instability of the initial configuration and the long‐time behavior is identified. The calculations further suggest that the corresponding continuous system will exhibit chaotic motions and energy‐sharing among a narrow band of unstable modes, a phenomenon which we call ‘‘confined chaos.’’

/pdf/instability-and-confined-chaos-in-a-nonlinear-dispersive-2ckd511e1b.pdf

Instability and confined chaos in a nonlinear dispersive wave system

Conditions are found for the appearance of non-uniform progressive waves of permanent form from a long-wave modulation of a finite-amplitude Stokes wave on deep water. The waveheight at which the modulated waves can occur is a very slowly decreasing function of the modulation wavelength for values up to 150 times the original wavelength. Some qualitative remarks are made about the problem of determining the stability of the new waves.

/pdf/long-wavelength-bifurcation-of-gravity-waves-on-deep-water-1ega9mhos1.pdf

Long wavelength bifurcation of gravity waves on deep water

Stern & Adam and subsequent workers have considered the linear stability of two-dimensional,
parallel, ideal fluid flow with shear in the presence of a free surface. In
these studies a fluid current is modelled as a finite layer of constant vorticity above
a semi-infinite stagnant region, corresponding to a piecewise-linear velocity profile.
Here, an investigation of the stability of currents for several smooth velocity profiles
is presented. With surface tension present it is found that the fluid surface velocity
must still exceed the minimum wavespeed of stagnant fluid for instability to occur;
a result highlighted by Caponi et al. for piecewise-linear profiles. Instability growth
rates are found to be significantly smaller than those associated with a piecewise-linear
profile. There are also qualitative differences in the stability characteristics; in
particular, transition is associated with an exchange of stability for smooth profiles,
but not for the piecewise-linear profile.

Waves generated by shear layer instabilities

Turbulence model predictions for flows with significant mean streamline curvature

Detailed comparisons are made between long-time numerical integration of the motion of four identical point vortices obtained using both a fourth-order symplectic integration method of the implicit Runge-Kutta type and a standard fourth-order explicit Runge-Kutta scheme. We utilize the reduced hamiltonian formulation of the four-vortex problem due to Aref & Pomphrey. Initial conditions which give both fully chaotic and also quasi-periodic motions are considered over integration times of order 10^6-10^7 times the characteristic time scale of the evolution. The convergence, as the integration time step is decreased, of the Poincare section is investigated. When smoothness of the section compared to the converged image, and the fractional change in the hamiltonian H are used as diagnostic indicators, it is found that the symplectic scheme gives substantially superior performance over the explicit scheme, and exhibits only an apparent qualitative degrading in results up to integration time steps of order the minimum timescale of the evolution. It is concluded that this performance derives from the symplectic rather than the implicit character of the method.

Philip Geoffrey Saffman

Papers

Instability and confined chaos in a nonlinear dispersive wave system

Long wavelength bifurcation of gravity waves on deep water

Waves generated by shear layer instabilities

Turbulence model predictions for flows with significant mean streamline curvature

Long-Time Symplectic Integration: The Example of Four-Vortex Motion