A Useful Approximation to the Error Function: Applications to Mass, Momentum, and Energy Transport in Shear Layers

1989 ◽  
Vol 111 (2) ◽  
pp. 224-226 ◽  
Author(s):  
P. R. Greene
Keyword(s):  
Closed Form ◽  
Energy Flux ◽  
Energy Transport ◽  
Error Function ◽  
Gaussian Function ◽  
Inverse Function ◽  
Shear Layers ◽  
Turbulent Shear ◽  
Flux Transport ◽  
Transport Flux

Specific mass, momentum, and energy flux transport integrals are given by the sequence I1 = ∫erfc(x)dx, I2 = ∫erfc2(x)dx, and I3 = ∫erfc3(x)dx for the error function velocity distribution typical of some laminar and turbulent shear layers. In this report, the Gaussian function A exp(−b(x−x0)2) is used to approximate the error function within 0.21 percent, allowing a direct approximate closed-form evaluation of these transport flux integrals. Mass, momentum, and energy flux are accurate to within 0.22, 0.15, and 0.11 percent, respectively, over the entire shear layer. To achieve this same degree of accuracy with a Taylor series requires in excess of 10 terms. Each of the sequence of approximate functions can be inverted to yield the inverse function, i.e., erfc−1(x), etc., also in closed-form. The results presented here are also applicable to the error function as it appears in heat transfer and probability and statistics type problems. A coefficient table is included to allow evaluation of the error function and its various integrals.

Free Turbulent Shear Layers on Plug Nozzles

1977 ◽  
Vol 99 (2) ◽  
pp. 301-308
Author(s):  
C. J. Scott ◽  
D. R. Rask
Keyword(s):  
Turbulent Mixing ◽  
Schmidt Number ◽  
Error Function ◽  
Mixing Layer ◽  
Shear Layers ◽  
Turbulent Shear ◽  
Gas Chromatograph ◽  
Plug Nozzle ◽  
Turbulent Schmidt Number ◽  
Plug Nozzles

Two-dimensional, free, turbulent mixing between a uniform stream and a cavity flow is investigated experimentally in a plug nozzle, a geometry that generates idealized mixing layer conditions. Upstream viscous layer effects are minimized through the use of a sharp-expansion plug nozzle. Experimental velocity profiles exhibit close agreement with both similarity analyses and with error function predictions. Refrigerant-12 was injected into the cavity and concentration profiles were obtained using a gas chromatograph. Spreading factors for momentum and mass were determined. Two methods are presented to determine the average turbulent Schmidt number. The relation Sct = Sc is suggested by the data for Sc < 2.0.

scholarly journals High-latitude electromagnetic and particle energy flux during an event with sustained strongly northward IMF

2005 ◽  
Vol 23 (4) ◽  
pp. 1295-1310 ◽  
Author(s):  
H. Korth ◽  
B. J. Anderson ◽  
H. U. Frey ◽  
C. L. Waters
Keyword(s):  
Magnetic Field ◽  
Solar Wind ◽  
Energy Flux ◽  
High Latitude ◽  
Energy Transport ◽  
Small Region ◽  
Radial Component ◽  
Particle Energy ◽  
Electromagnetic Energy ◽  
Viscous Interaction

Abstract. We present a case study of a prolonged interval of strongly northward orientation of the interplanetary magnetic field on 16 July 2000, 16:00-19:00 UT to characterize the energy exchange between the magnetosphere and ionosphere for conditions associated with minimum solar wind-magnetosphere coupling. With reconnection occurring tailward of the cusp under northward IMF conditions, the reconnection dynamo should be separated from the viscous dynamo, presumably driven by the Kelvin-Helmholtz (KH) instability. Thus, these conditions are also ideal for evaluating the contribution of a viscous interaction to the coupling process. We derive the two-dimensional distribution of the Poynting vector radial component in the northern sunlit polar ionosphere from magnetic field observations by the constellation of Iridium satellites together with drift meter and magnetometer observations from the Defense Meteorological Satellite Program (DMSP) F13 and F15 satellites. The electromagnetic energy flux is then compared with the particle energy flux obtained from auroral images taken by the far-ultraviolet (FUV) instrument on the Imager for Magnetopause to Aurora Global Exploration (IMAGE) spacecraft. The electromagnetic energy input to the ionosphere of 51 GW calculated from the Iridium/DMSP observations is eight times larger than the 6 GW due to particle precipitation all poleward of 78° MLAT. This result indicates that the energy transport is significant, particularly as it is concentrated in a small region near the magnetic pole, even under conditions traditionally considered to be quiet and is dominated by the electromagnetic flux. We estimate the contributions of the high and mid-latitude dynamos to both the Birkeland currents and electric potentials finding that high-latitude reconnection accounts for 0.8 MA and 45kV while we attribute <0.2MA and ~5kV to an interaction at lower latitudes having the sense of a viscous interaction. Given that these conditions are ideal for the occurrence of the KH instability at the magnetopause and hence the viscous interaction, this result suggests that the viscous interaction is a small contributor to coupling solar wind energy to the magnetosphere-ionosphere system.

Management of Turbulent Shear Layers in Separated Flow

1978 ◽  
Vol 15 (7) ◽  
pp. 385-386 ◽  
Author(s):  
W.W. Willmarth ◽  
R.F. Gasparovic ◽  
J.M. Maszatics ◽  
J.L. McNaughton ◽  
D.J. Thomas
Keyword(s):  
Separated Flow ◽  
Shear Layers ◽  
Turbulent Shear

Mechanisms of Forced Tropical Meridional Energy Flux Change

2015 ◽  
Vol 28 (5) ◽  
pp. 1725-1742 ◽  
Author(s):  
Spencer A. Hill ◽  
Yi Ming ◽  
Isaac M. Held
Keyword(s):  
Spatial Pattern ◽  
Greenhouse Gas ◽  
Energy Flux ◽  
General Circulation ◽  
Energy Transport ◽  
Circulation Model ◽  
Hadley Cell ◽  
Energy Fluxes ◽  
Hadley Cells ◽  
Sst Anomalies

Abstract Anthropogenically forced changes to the mean and spatial pattern of sea surface temperatures (SSTs) alter tropical atmospheric meridional energy transport throughout the seasonal cycle—in total, its partitioning between the Hadley cells and eddies and, for the Hadley cells, the relative roles of the mass flux and the gross moist stability (GMS). The authors investigate this behavior using an atmospheric general circulation model forced with SST anomalies caused by either historical greenhouse gas or aerosol forcing, dividing the SST anomalies into two components: the tropical mean SST anomaly applied uniformly and the full SST anomalies minus the tropical mean. For greenhouse gases, the polar-amplified SST spatial pattern partially negates enhanced eddy poleward energy transport driven by mean warming. Both SST components weaken winter Hadley cell circulation and alter GMS. The Northern Hemisphere–focused aerosol cooling induces northward energy flux anomalies in the deep tropics, which manifest partially via strengthened northern and weakened southern Hadley cell overturning. Aerosol-induced GMS changes also contribute to the northward energy fluxes. A simple thermodynamic scaling qualitatively captures these changes, although it performs less well for the greenhouse gas simulations. The scaling provides an explanation for the tight correlation demonstrated in previous studies between shifts in the intertropical convergence zone and cross-equatorial energy fluxes.

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