Fluid dynamics Reynolds numbers

Voinov, O. V. and Petrov, A. G., Flows with closed lines of flow and motion of droplets at high Reynolds numbers, Fluid Dynamics, Vol. 22, No. 5, pp. 708-717, 1987. 

Koehl MAR (1992) Hairy little legs feeding, smelling, and swimming at low Reynolds number. Fluid dynamics in biology. Contemp Math 141 33-64... 

Rivkind, V. Y. and Ryskin, G. M. Flow structure in motion of a spherical chop in a fluid medium at intermediate Reynolds numbers. Fluid Dynamics (English translation of Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza), 11, 5-12,1976. 

G5b. Goldman, A. J., Cox, R. G., and Brenner, H., Slow viscous motion of two identical arbitrarily oriented spheres through a viscous fluid. Chem. Eng. Sci. (in press) see also Goldman, A. J., Investigations in low Reynolds number fluid-particle dynamics. Ph.D. Dissertation, New York University, New York, 1966. 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

The Reynolds number for flow in a tube is defined by dvpirj, where d is the diameter of the tube, V is the average velocity of the fluid along the tube, p is the density of the fluid, and rj is its dynamic viscosity. At flow velocities corresponding with values of the Reynolds number of greater than 2000, turbulence is encountered. 

A numerical study of the effect of area ratio on the flow distribution in parallel flow manifolds used in a Hquid cooling module for electronic packaging demonstrate the useflilness of such a computational fluid dynamic code. The manifolds have rectangular headers and channels divided with thin baffles, as shown in Figure 12. Because the flow is laminar in small heat exchangers designed for electronic packaging or biochemical process, the inlet Reynolds numbers of 5, 50, and 250 were used for three different area ratio cases, ie, AR = 4, 8, and 16. 

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

For an incompressible viscous fluid (such as the atmosphere) there are two types of flow behaviour 1) Laminar, in which the flow is uniform and regular, and 2) Turbulent, which is characterized by dynamic mixing with random subflows referred to as turbulent eddies. Which of these two flow types occurs depends on the ratio of the strengths of two types of forces governing the motion lossless inertial forces and dissipative viscous forces. The ratio is characterized by the dimensionless Reynolds number Re. 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

The dynamical regimes that may be explored using this method have been described by considering the range of dimensionless numbers, such as the Reynolds number, Schmidt number, Peclet number, and the dimensionless mean free path, which are accessible in simulations. With such knowledge one may map MPC dynamics onto the dynamics of real systems or explore systems with similar characteristics. The applications of MPC dynamics to studies of fluid flow and polymeric, colloidal, and reacting systems have confirmed its utility. 

M. Ripoll, K. Mussawisade, R. G. Winkler, and G. Gompper, Low-Reynolds-number hydrodynamics of complex fluids by multi-particle-collision dynamics, Europhys. Lett. 68, 106... 

This is valid for any Newtonian fluid in any (circular) pipe of any size (scale) under given dynamic conditions (e.g., laminar or turbulent). Thus, if the values of jV3 (i.e., the Reynolds number 7VRe) and /V, (e/D) for an experimental model are identical to the values for a full-scale system, it follows that the value of N6 (the friction factor) must also be the same in the two systems. In such a case the model is said to be dynamically similar to the full-scale (field) system, and measurements of the variables in N6 can be translated (scaled) directly from the model to the field system. In other words, the equality between the groups /V3 (7VRc) and N (e/D) in the model and in the field is a necessary condition for the dynamic similarity of the two systems. 

The CFD model described above has been used by Liu and Fox (2006) to simulate the experiments of Johnson and Prud homme (2003a) in a confined impinging-jets reactor. In these experiments, two coaxial impinging jets with equal flow rates are used to introduce the two reactant-streams. The jet Reynolds number Re, determines the fluid dynamics in the reactor. Typical CFD results are shown in Fig. 6 9 for a jet Reynolds number of Re, = 400 and a reaction time of tr — 4.8 msec. The latter is controlled by fixing the inlet concentrations of the reactants. Further, details on the reactor geometry and the CFD model can be found in Liu and Fox (2006). 

In equation 1.3, p is the density, p. the dynamic viscosity, and the mean velocity of the fluid d, is the inside diameter of the tube. Any consistent system of units can be used in this equation. The Reynolds number is also frequently written in the form... 

Generally, the flow field is assumed as a laminar flow, due to the relatively low velocities that the air reaches inside the cell. Nevertheless, Campanari and Iora (2004) performed a fluid dynamic calculation of the flow in the air injection tube and in the annular section of the cell the results indicated a transition from laminar to turbulent flow the values of the Reynolds number found were in some cases above 1000, whereas the transition between laminar and turbulent flow is stated to be in the range between Re = 750 and Re = 2700. The regime of the flow affects the heat exchange between the gas and the solid material and the diffusion of chemical species. Li and Suzuki (2004) too performed similar calculations and found values of the Reynolds number that were consistent with a regime transition in the air injection tube, but not for the annular section (Re = 385 with a velocity lower than 7.82 m/s). Li and Chyu (2003) state that the assumption of laminar flow is to be rejected. Other researchers, such as Haynes and Wepfer (2001) previously and Stiller et al. (2005) later, assume laminar flow. 

To consolidate the experimental screening data quantitatively it is desirable to obtain information on the fluid mechanics of the reactant flow in the reactor. Experimental data are difficult to evaluate if the experimental conditions and, especially, the fluid dynamic behavior of the reactants flow are not known. This is, for example, the case in a typical tubular reactor filled with a packed bed of porous beads. The porosity of the beads in combination with the unknown flow of the reactants around the beads makes it difficult to describe the flow close to the catalyst surface. A way to achieve a well-described flow in the reactor is to reduce its dimensions. This reduces the Reynolds number to a region of laminar flow conditions, which can be described analytically. 

In Damkohler s analysis, which applied to a continuous chemical reaction process in a tubular reactor, he solved these dilemmas by completely abandoning geometric similarity and fluid dynamic similarity. In other words, L/D idem and assuming that the Reynolds number is irrelevant in the scaling. Hence, his scale-up depends exclusively on thermal and reaction similarity. In our case it is even easier to see that the Reynolds number is very small and does not play a role in the process. By allowing to adjust L/D accordingly, there is more flexibility in the scaling problem. 

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