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.

/pdf/convection-heat-transfer-47drgw6czo.pdf

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

Steady, inviscid, incompressible, two-dimensional flows with vortex patches bounded by vortex sheets (Batchelor flows) are calculated numerically. Two particular cases are considered: the vortex on a plane wall (Sadovskii vortex) and the vortex in a right-angled corner. Nonlinear integral equations are derived for the shape of the bounding vortex sheet which are solved numerically. Two different formulations are employed to check the results. Previous results by Sadovskii [Appl. Math. Mech. 35, 773 (1971)] and Chernyshenko (Royal Aircraft Establishment library translations Report No. 2133, 1983) for specific values of the parameters are confirmed. Only symmetrical solutions are found to exist.

/pdf/the-calculation-of-some-batchelor-flows-the-sadovskii-vortex-515ncgbx0s.pdf

The calculation of some Batchelor flows: The Sadovskii vortex and rotational corner flow

Solutions for interfacial waves of permanent form in the presence of a current wcre obtained for small-to-moderate wave amplitudes. A weakly nonlinear approximation was used to give simple analytical solutions to second order in wave height. Numerical methods were usctl to obtain solutions for larger wave amplitudes, details are reported for a number of selected cases. A special class of finite-amplitude solutions, closely related to the well-known Stokes surface waves, were identified. Factors limiting the existence of steady solutions are examined.

/pdf/finite-amplitude-interfacial-waves-in-the-presence-of-a-4u0y21c1g4.pdf

Finite-amplitude interfacial waves in the presence of a current

The movement of a horizontal vortex pair through an inhomogeneous fluid is considered. The problem is formulated first for the case when the ambient fluid is uniform, the fluid moving with the vortex pair has a different density, and the motion is supposed laminar and inviscid. An approximate solution is obtained, which predicts that the distance between the vortices stays constant and the vortices accelerate at a constant rate. This solution is then applied to motion in a stratified atmosphere and it is found that the vortices oscillate vertically with a frequency and amplitude depending on the initial conditions and the stratification. Finally, approximate equations are constructed to describe the effects of turbulent entrainment into the fluid moving with the vortex pair, and an estimate of the damping is obtained.

The Motion of a Vortex Pair in a Stratified Atmosphere

The stability of two-dimensional infinitesimal disturbances of the steady inviscid incompressible flow produced by infinite rows of finite-cored particles is studied. Numerical results are obtained for the growth rate and oscillation frequencies of disturbances of arbitrary subharmonic wavenumber, and the stability boundaries are calculated. The stabilization of the pairing instability by finite area demonstrated by Saffman and Schatzman (1982) is confirmed, as is Kida's (1982) result that this is not the most unstable disturbance when the area is finite. However, the finite area does not stabilize the inviscid Karman vortex street of finite-area vortices to infinitesimal two-dimensional disturbances of arbitrary wavelength. The finite area is always unstable except for one isolated value of the aspect ratio which depends upon the size of the vortices.

/pdf/the-linear-two-dimensional-stability-of-inviscid-vortex-1rujmz5w84.pdf

The linear two-dimensional stability of inviscid vortex streets of finite-cored vortices

An inviscid model for the Karman vortex street, containing vortices of uniform vorticity surrounded by irrotational fluid, is related to the wake behind a bluff body by a global analysis requiring the conservation of momentum, energy and vorticity. Some comparison is made with experimental results reported in the literature. A qualitative procedure is proposed whereby the slow evolution of the wake through viscous effects is approximated. Some comments are made regarding the relevance of the stability properties of the inviscid street. Some calculations are made for the ‘secondary vortex street’ that is observed after breakdown and rearrangement, and comparison is made with experiment.

/pdf/an-inviscid-model-for-the-vortex-street-wake-4hx69po3xq.pdf

Philip Geoffrey Saffman

Papers

The calculation of some Batchelor flows: The Sadovskii vortex and rotational corner flow

Finite-amplitude interfacial waves in the presence of a current

The Motion of a Vortex Pair in a Stratified Atmosphere

The linear two-dimensional stability of inviscid vortex streets of finite-cored vortices

An inviscid model for the vortex-street wake