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

This paper is concerned with the dispersion of a dynamically neutral material quantity in a fluid flowing throuh a porous medium. The medium is regarded as an assemblage of randomly orientated straight pores, and it is assumed that the path of a marked element of the material quantity consists of a sequence of statistically independent steps whose direction and duration vary in some random manner. The probability density function for the displacement of a single marked element is calculated and values for the dispersion of a cloud of marked elements then follow.The case is examined in which the flow satisfies Darcy's law (i.e. the mean velocity is linearly proportional to the mean pressure gradient), and the molecular diffusivity is sufficiently small for the dispersion to be primarily due to the randomness of the streamlines, but it is not assumed that effects of molecular diffusion can be altogether neglected. It is shown that the longitudinal dispersion in the direction of the mean flow may be described asymptotically by an effective diffusivity which is a function of U, l, a, K and T. (U denotes the average velocity, l the pore length, a the pore radius which is shown to be related to the permeability, k the molecular diffusivity, and T the time from the initial instant.) Expressions for the longitudinal diffusivity kl are obtained according to the relative values of l/U, T, t0 = l2/2k and t1 = a2/8k. These are given in §4, equations (4.3), (4.4) and (4.5). Speaking roughly, when t0 [Gt ] T [Gt ] l/U,kl/Ul is alogarithmic function of UT/l and increases with T; when T [Gt ] t0 [Gt ] l/U, which must eventually be the case however small k,kl/Ul is alogarithmic function of Ul/k and independent of T. The theoretical results are compared with experimental data reported in the literature and approximate agreement is obtained when E is put equal to the average diameter of the particles composing the porous medium.The lateral dispersion in the direction perpendicular to that of the mean velocity is found to be governed asymptotically by an effective diffusivity . However, it is pointed out that some of the assumptions, namely that successive steps are statistically independent and that the dispersion of a cloud follows immediately from the statistical properties of the displacement of a single marked element, may not be valid for the lateral dispersion and this result is therefore suspect.Remarks are made in §5 on the dispersion for high values of the Reynolds number Ul/v (v = kinematic viscosity) when Darcy's law is not obeyed, and it is argued that kl/Ul should decrease as Ul/v increases.

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A theory of dispersion in a porous medium

The drag on a cylindrical particle moving in a thin sheet of viscous fluid is calculated. It is supposed that the sheet is embedded in fluid of much lower viscosity. A finite steady drag is obtained, which depends logarithmically on the ratio of the viscosities. The Einstein relation is used to determine the diffusion coefficient for Brownian motion of the particle, with application to the movement of molecules in biological membranes. In addition, the Brownian motion is calculated using the Langevin equation, and a logarithmically time-dependent diffusivity is obtained for the case when the embedding fluid has zero viscosity.

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Brownian motion in thin sheets of viscous fluid

A field of homogeneous turbulence generated at an initial instant by a distribution of random impulsive forces is considered. The statistical properties of the forces are assumed to be such that the integral moments of the cumulants of the force system all exist. The motion generated has the property that at the initial instant E(kappa) = Ckappa^2 + o(kappa^2) where E(k) is the energy spectrum function, k is the wave-number magnitude, and C is a positive number which is not in general zero. The corresponding forms of the velocity covariance spectral tensor and correlation tensor are determined. It is found that the terms in the velocity covariance Rij(r) are O(r^−3) for large values of the separation magnitude r. 

An argument based on the conservation of momentum is used to show that C is a dynamical invariant and that the forms of the velocity covariance at large separation and the spectral tensor at small wave number are likewise invariant. For isotropic turbulence, the Loitsianski integral diverges but the integral \[ \int_0^{\infty} r^2R(r)dr = \frac{1}{2}\pi C \] exists and is invariant.

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The large-scale structure of homogeneous turbulence

The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal, structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.

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Viscous fingering in Hele-Shaw cells

The motion of vortex rings of small cross section is considered. A formula is given for the velocity of a ring in an ideal fluid with an arbitrary distribution of vorticity in the core and an arbitrary circumferential velocity. A definition of velocity is given for the unsteady diffusing ring in a viscous fluid, and the speed is found by the method used for rings in ideal fluids.

Philip Geoffrey Saffman

Papers

A theory of dispersion in a porous medium

Brownian motion in thin sheets of viscous fluid

The large-scale structure of homogeneous turbulence

Viscous fingering in Hele-Shaw cells

The Velocity of Viscous Vortex Rings