Reynolds Number Scaling

In order to estimate an upper bound for attainable TZ s, consider a LG with lattice spacing Iq, speed-of-sound Cg, kinematic viscosity v. Reynold s number is then given by 

We can also estimate the required computational work. Since the evolution time of the LG is of order Ljv and the time required to update a single step on the lattice is of order Iq/cs, the LG simulation requires roughly L/[IqM) = TZ/M time steps. Assuming that the computational work for each site is of order 1, we have that the LG simulation therefore requires a memory storage space of order S (TZ/M) and a computational work requirement of order W 7 d+l/ d+2 

These values can be compared to predicted values for numerical solutions of the incompressible Navier-Stokes equations. For d = 2, for example, we have the lower bounds Sa=2 (TZ/M) and Wd=2 TZ /M for a LG and the bounds S num, d=2 and d=2 where the bounds for the numerical solutions 

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Equation (3.43) can be used to determine that the pressure scaling factor is o.86 Reynolds number scales as S/Sr =... 

Using these Reynolds number scale factors, the errors in the dimensionless drag coefficient j3L/psua using the simplified scaling models are shown in Figs. 24 and 25 for u0/umf of 10 and 1000, respectively, plotted as a function of Rep, based on parameters for the exact scaled bed. For a particle Reynolds number of 1000 or less, which corresponds to... 

For non-Newtonian liquids and suspensions, an apparent viscosity is determined using correlations which include power input and the Reynolds number. Scale-up comparisons based on heat generation data only were determined by comparison of results from RC1 experiments and from a 675-liter reactor [208]. In the experiments, a Bingham plastic fluid was used to determine the film heat transfer coefficient. This presents a worst case because of the low thermal conductivity of the Bingham plastic. Calculated inside film heat transfer coefficients determined in the RC1 tests were about 60% lower than the values determined in the pilot plant reactor, even though substantial effort was made to obtain both geometric and kinematic similarity in the pilot reactor. 

DeGraaff, D. B. and J. K. Eaton. Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Lluid Mech. 422, 319-346 (2000). 

Recall that there is a fundamental scaling difference between the cylindrical wedge flow and the spherical inclined-disk flow. In the wedge flow, the Reynolds number is independent of r, whereas in the spherical case, the Reynolds number scales as /r. Thus, in the spherical case, there is a different Reynolds number at every radial position in the channel. In practice, a quantitative determination of the velocity profile is more complex in the spherical case. The nondimensional velocity profile must be determined at each radial position where the actual velocity profile is desired. 

Yoona H.S., Hillb D.F., Balachandra S., Adrianc R.J., Haa M.Y Reynolds number scaling of flowing a Ruston turbine stirred tank. Mean flow, circular jet and tip vortex scaling. Chemical Biochemical Engineering, New Brunswick, Rutgers University, 2004. 

ColUns, L.R. (2000). Reynolds Number Scaling of Preferential Concentration of Particles in Isotropic Turbulence. AIChE Aimual Meeting, Los Angeles, CA. 

The potential lack of fnlly turbulent flow (failure of Reynolds number scaling) in regions distant from the impeller... 

This is set to 1. A constant residence time is imposed by setting S Sl = S. There are two equations and two unknowns, Sr and Sl- The solution is Sr = and Sl = S The length-to-diameter ratio scales as S . Equation (3.43) can be used to determine that the pressure scaling factor is o.86 pjjg Reynolds number scales as S/Sr =... 

Collins LR, Keswani A Reynolds number scaling of particle clustering in turbulent aerosols. New J Phys 6 119, 2004. http //dx.doi.Org/10.1088/1367-2630/6/l/119. 

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