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The collision cross section per volume or collision cross section density is , and it is related to the mean free path by l = 1 2 C σ {\displaystyle l={\frac {1}{{\sqrt {2}}C\sigma }}} Combining the kinetic equations for molecular motion with the defining equation of shear viscosity gives the well known equation for shear viscosity for dilute ...
V is the cross-sectional average velocity (dimension of L/T; units of ft/s or m/s); n is the Gauckler–Manning coefficient. Units of n are often omitted, however n is not dimensionless, having dimension of T/L 1/3 and units of s/m 1/3. R h is the hydraulic radius (L; ft, m);
Large eddy simulation of a turbulent gas velocity field.. Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics.It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, [1] and first explored by Deardorff (1970). [2]
Mathematically, mass flux is defined as the limit =, where = = is the mass current (flow of mass m per unit time t) and A is the area through which the mass flows.. For mass flux as a vector j m, the surface integral of it over a surface S, followed by an integral over the time duration t 1 to t 2, gives the total amount of mass flowing through the surface in that time (t 2 − t 1): = ^.
The average kinetic energy then involves the root mean-square velocity, which always exceeds the mean velocity. In the case of turbulent flow , the fluid acquires random velocity components in all directions, including perpendicular to the length of the pipe, and thus turbulence contributes to the kinetic energy per unit volume but not to the ...
For pipe flows a so-called transit time method is applied where a radiotracer is injected as a pulse into the measured flow. The transit time is defined with the help of radiation detectors placed on the outside of the pipe. The volume flow is obtained by multiplying the measured average fluid flow velocity by the inner pipe cross-section.
Favre averaging is the density-weighted averaging method, used in variable density or compressible turbulent flows, in place of the Reynolds averaging.The method was introduced formally by the French physicist Alexandre Favre in 1965, [1] [2] although Osborne Reynolds had also already introduced the density-weighted averaging in 1895. [3]
where is the volume fraction of the fibers in the composite (and is the volume fraction of the matrix).. If it is assumed that the composite material behaves as a linear-elastic material, i.e., abiding Hooke's law = for some elastic modulus of the composite and some strain of the composite , then equations 1 and 2 can be combined to give