Skin friction in zero-pressure-gradient boundary layers

A global approach leading to a self-consistent solution to the Navier-Stokes-Prandtl equations for zero-pressure-gradient boundary layers is presented. It is shown that as $Re_{\delta}\rightarrow \infty$, the dynamically defined boundary layer thickness $\delta(x)\propto x/\ln^{2}Re_{x}$ and the skin friction $\lambda=\frac{2\tau_{w}}{\rho U_{0}^{2}}\propto 1/\ln^{2}\delta(x)$. Here $\tau_{w}$ and $U_{0}$ are the wall shear stress and free stream velocity, respectively. The theory is formulated as an expansion in powers of a small dimensionless parameter $\frac{d\delta(x)}{dx}\rightarrow 0$ in the limit $x\rightarrow \infty$

The Last Stages of Terrestrial Planet Formation: Dynamical Friction and the Late Veneer

The final stage of terrestrial planet formation consists of the cleanup of residual planetesimals after the giant impact phase. Dynamically, a residual planetesimal population is needed to damp the high eccentricities of the terrestrial planets after the giant impact stage. Geochemically, highly siderophile element (HSE) abundance patterns inferred for the terrestrial planets and the Moon suggest that a total of about 0.01 M_Earth of chondritic material was delivered as `late veneer' by planetesimals to the terrestrial planets after the end of giant impacts. Here we combine these two independent lines of evidence for a leftover population of planetesimals and show that: 1) A residual planetesimal population containing 0.01 M_Earth is able to damp the eccentricities of the terrestrial planets after giant impacts to their observed values. 2) At the same time, this planetesimal population can account for the observed relative amounts of late veneer added to the Earth, Moon and Mars provided that the majority of the late veneer was delivered by small planetesimals with radii <10m. These small planetesimal sizes are required to ensure efficient damping of the planetesimal's velocity dispersion by mutual collisions, which in turn ensures that the planets' accretion cross sections are significantly enhanced by gravitational focusing above their geometric values. Specifically we find, in the limit that the relative velocity between the terrestrial planets and the planetesimals is significantly less than the terrestrial planets' escape velocities, that gravitational focusing yields an accretion ratio Earth/Mars~17, which agrees well with the accretion ratio inferred from HSEs of 12-23. For the Earth-Moon system, we find an accretion ratio of ~200, which is consistent with estimates of 150-700 derived from HSE abundances that include the lunar crust as well as mantle component. (Abridged)Comment: accepted for publication in ApJ, 9 pages, 4 figures; minor corrections, additional references adde

Experimental Investigation of the Problem of Surface Roughness

Based on the universal laws of turbulent velocity distribution at rough and smooth walls, there is in the present work presented a method that allows surface roughness tests and in particular, measurements on the roughness of ship surfaces to be carried out in a much simpler manner. The types of roughness investigated were in the form of flat, rough plates installed in a square-section rectangular channel, the other three walls always being smooth. Twenty-one plates of various roughness were investigated, the roughness elements being the following: spheres of diameter 0.41 and 0.21, respectively, spherical segments, cones, and "short" and "long" angles

Airfoil Theory at Supersonic Speed

A theory is developed for the airfoil of finite span at supersonic speed analogous to the Prandtl airfoil theory of 1918-1919 for incompressible flow. In addition to the profile and induced drags, account must be taken at supersonic flow of still another drag, namely, the wave drag, which is independent of the wing aspect ratio. Both wave and induced drags are proportional to the square of the lift and depend on the Mach number, that is, the ratio of flight to sound speed. In general, in the case of supersonic flow, the drag-lift ratio is considerably less favorable than is the case for incompressible flow. Among others the following examples are considered: 1) lifting line with constant lift distribution (horseshoe vortex); 2) computation of wave and induced drag and the twist of a trapezoidal wing of constant lift density; 3) computation of the lift distribution and drag of an untwisted rectangular wing

Amplitude distribution and energy balance of small disturbances in plate flow

The distribution of the correlation coefficient and of the amplitude of the disturbance velocities is calculated as a function of the distance from the wall for two neutral disturbances, one at the lower and one at the upper branch of the neutral stability curve. The energy balance of the disturbance motion is also investigated and it is found that as required for neutral stability the energy of the disturbance motion that is dissipated by viscosity is equal to the energy transferred to the disturbance motion from the main flow during one cycle

The Dornier Wind Tunnel

After completion of the required calibrations, the Dornier open-throat tunnel is now in operation. With an elliptic test section of 3 by 4 m (9.84 by 3.12 ft.), its length is 7 m (22.97 ft.), its maximum horsepower 800, and its maximum air speed 60 m/s (134.2 mph). As to local uniformity of velocity, static pressure as well as jet direction, and turbulence factor, this tunnel is on par with those of the good German and foreign research labs

Is turbulent mixing a self convolution process ?

Experimental results for the evolution of the probability distribution function (PDF) of a scalar mixed by a turbulence flow in a channel are presented. The sequence of PDF from an initial skewed distribution to a sharp Gaussian is found to be non universal. The route toward homogeneization depends on the ratio between the cross sections of the dye injector and the channel. In link with this observation, advantages, shortcomings and applicability of models for the PDF evolution based on a self-convolution mechanisms are discussed.Comment: 4 page

Integral analysis of laminar indirect free convection boundary layers with weak blowing for Schmidt no. ~ 1

Laminar natural convection at unity Schmidt number over a horizontal surface with a weak normal velocity at the wall is studied using an integral analysis. To characterise the strength of the blowing, we define a non-dimensional parameter called the blowing parameter. After benchmarking with the no blowing case, the effect of the blowing parameter on boundary layer thickness, velocity and concentration profiles is obtained. Weak blowing is seen to increase the wall shear stress. For blowing parameters greater than unity, the diffusional flux at the wall becomes negligible and the flux is almost entirely due to the blowing.Comment: 10 pages, published in International Communications in heat and mass transfer,Vol31,No8, 2004, pp 1199 -120

Morphological instability of the solid-liquid interface in crystal growth under supercooled liquid film flow and natural convection airflow

Ring-like ripples on the surface of icicles are an example of morphological instability of the ice-water interface during ice growth under supercooled water film flow. The surface of icicles is typically covered with ripples of about 1 cm in wavelength, and the wavelength appears to be almost independent of external temperature, icicle radius, and volumetric water flow rate. One side of the water layer consists of the water-air surface and growing ice is the other. This is one of the more complicated moving phase boundary problems with two interfaces. A recent theoretical work [K. Ueno, Phys. Rev. E 68, (2003) 021603] to address the underlying instability that produces ripples is based on the assumption of the absence of airflow around icicles. In this paper, we extend the previous theoretical framework to include a natural convection airflow ahead of the water-air surface and consider whether the effect of natural convection airflow on the wavelength of ripples produced on an ice surface is essential or not.Comment: 19 pages, 5 figure