Showing papers on "Similarity solution published in 1968"

Lagrangian‐History Statistical Theory for Burgers' Equation

Abstract: The Lagrangian‐history direct‐interaction approximation for Burgers' equation yields a k−2 inertialrange spectrum and an infinite‐Reynolds‐number similarity solution which describes stationary, decaying sawtooth shock waves. This solution differs by a numerical factor from an exact statistical solution of Burgers' equation into which almost all spatially periodic, zero‐mean, infinite‐Reynolds‐number initial ensembles are expected to evolve. The different inertial‐range predictions of the approximation for Burgers' equation and Navier‐Stokes dynamics (where k−5/3 results) are directly associated with the effects of pressure‐induced accelerations on Lagrangian correlation times.

Laminar natural convection about an isothermally heated sphere at small Grashof number

Abstract: The flow induced by gravity about a very small heated isothermal sphere introduced into a fluid in hydrostatic equilibrium is studied. The natural-convection flow is taken to be steady and laminar. The conditions under which the Boussinesq model is a good approximation to the full conservation laws are described. For a concentric finite cold outer sphere with radius, in ratio to the heated sphere radius, roughly less than the Grashof number to the minus one-half power, a recirculating flow occurs; fluid rises near the inner sphere and falls near the outer sphere. For a small heated sphere in an unbounded medium an ordinary perturbation expansion essentially in the Grashof number leads to unbounded velocities far from the sphere; this singularity is the natural-convection analogue of the Whitehead paradox arising in three-dimensional low-Reynolds-number forced-convection flows. Inner-and-outer matched asymptotic expansions reveal the importance of convective transport away from the sphere, although diffusive transport is dominant near the sphere. Approximate solution is given to the nonlinear outer equations, first by seeking a similarity solution (in paraboloidal co-ordinates) for a point heat source valid far from the point source, and then by linearization in the manner of Oseen. The Oseen solution is matched to the inner diffusive solution. Both outer solutions describe a paraboloidal wake above the sphere within which the enthalpy decays slowly relative to the rapid decay outside the wake. The updraft above the sphere is reduced from unbounded growth with distance from the sphere to constant magnitude by restoration of the convective accelerations. Finally, the role of vertical stratification of the ambient density in eventually stagnating updrafts predicted on the basis of a constant-density atmosphere is discussed.

The slow motion of a flat plate in a viscous stratified fluid

Abstract: The existence of a ‘wake’ upstream of an obstacle moving slowly through a stratified fluid has been known for some time. The present study shows that a thin, flat plate moving slowly and horizontally through a linearly stratified salt-water mixture has, in addition, a boundary layer over the plate whose thickness increases upstream from the back of the plate.The theory assumes that the ratio of diffusivity to viscosity is small, and that the plate moves so slowly that inertia forces are negligible; under these conditions, a similarity solution is derived describing the boundary layer over the plate. The study also shows that salt diffusion is important in a second, thinner boundary layer whose thickness increases from the front of the plate.In the experiment, a plate was towed through a tank of linearly stratified salt water. From streak photographs of the boundary layer over the plate, it was possible to confirm quantitatively the similarity solution and to infer at very slow velocities the presence of the thin diffusion boundary layer.

Free convection from a flat plate

Abstract: A numerical solution is presented for the development of free convection from a semi-infinite vertical flat plate which is uniformly heated up to a length l from the base and insulated for the rest of its length. At great heights above the heated part of the plate, the velocity and temperature distributions behave as if the heat were put in as a line source of heat at the base of the plate. Matching of the solutions for the heated and the insulated parts of the plate, by keeping the fluxes of heat and momentum continuous, determines the position of the effective origin of the similarity solution for the insulated plate in terms of the length, l, of the heated part of the plate. Graphs of the dimensionless velocity, temperature, heat flux and axial length parameters are given for different values of the Prandtl number.

Drag Reduction of a Non-Newtonian Fluid by Fluid Injection at the Wall

Abstract: The flat plate boundary-layer equations for a power law model of a non-Newtonian fluid are formulated in a manner to allow similarity solutions with mass injection at the boundary. The variation of injection velocity with longitudinal position along the plate which allows a similarity solution is determined as a function of the power law exponent N. In the analysis, it is shown that the two- point boundary value problem can be reduced to an equivalent initial value problem. Numerical results are presented for velocity profiles, skin-friction coefficient, displacement and momentum thicknesses for a range of values of the power law exponent, and the dimensionless mass injection parameter.

The laminar wall-jet over a curved surface

Abstract: The laminar flow of a wall jet over a curved surface is considered. A unique similarity solution is obtained for both concave and convex surfaces when the local radius of curvature is proportional to x3/4. This solution satisfies a similar invariant condition to the one derived by Glauert for the wall jet over a plane surface. The variation of the shape of the velocity profile, the skin friction, and the surface pressure as a function of curvature is given.

Similarity Solution for Cylindrical Gas Cloud in Rarefied Atmosphere

Abstract: The analysis of Stuart for an expanding spherical gas cloud in a rarefied atmosphere is extended to the case of cylindrical flow. The equations of motion are expressed in similarity form from which early and late time solutions are obtained. The spherical and cylindrical flows are numerically compared for the case of negligible frictional heating by the atmosphere.

Similarity solution for a laminar, incompressible jet flowing along a curved surface.

Abstract: Similarity transformation and perturbation solution for laminar two dimensional incompressible jet flow over curved surface of small curvature