Newton turbulent flows

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

The power input in stirred tanks can be calculated using the equation P = Ne pnM if the Newton number Ne, which at present still has to be determined by empirical means, is known. For stirred vessels with full reinforcement (bafQes, coils, see e.g. [20]), the only bioreactors of interest, this is a constant in the turbulent flow range Re = nd /v> 5000-10000, and in the non-aerated condition depends only on geometry (see e.g. [20]). In the aerated condition the Newton number is also influenced by the Froude number Fr = n d/g and the gas throughput number Q = q/nd (see e.g. [21-23]). 

This expression for the terminal velocity (i.e., the constant velocity that the particle ultimately attains), is called Stokes law. When the Reynolds number is high, say usually greater than of the order of 800, the flow becomes turbulent flow and eddies form. It was Newton... 

Likewise, Newton s law for terminal velocity (turbulent flow) can be expressed in simplified form as... 

In the Lagrangian approach, individual parcels or blobs of (miscible) fluid added via some feed pipe or otherwise are tracked, while they may exhibit properties (density, viscosity, concentrations, color, temperature, but also vorti-city) that distinguish them from the ambient fluid. Their path through the turbulent-flow field in response to the local advection and further local forces if applicable) is calculated by means of Newton s law, usually under the assumption of one-way coupling that these parcels do not affect the flow field. On their way through the tank, these parcels or blobs may mix or exchange mass and/or temperature with the ambient fluid or may adapt shape or internal velocity distributions in response to events in the surrounding fluid. 

In the range Re > 50 (vessel with baffles) or Re > 5 x 10" (unbaffled vessel), because the Newton number Ne = P/(pn d ) remains constant. In this case, viscosity is irrelevant we are dealing with a turbulent flow region. 

Eq. 3. 85 is Newton s law of viscosity. It is experimentally observed by several liquids provided their rate of flow is not very high. The fluid flow to which this equation applies is called laminar (or streamlined) flow. The equation does not apply to turbulent flow. Liquids which do not obey Eq. 3.85 are called non-Newtonian liquids. For Newtonian liquids, 77 is independent of dv/dxwhereas for non-Newtonian liquids, 77 changes as dv/dz changes. 

Gasljevic, K. Aguilar, G. Matthys, E.F. An improved diameter scaling correlation for turbulent flow of drag-reducing polymer solutions. J. Non-Newton. Fluid Mech. 1999, 84, 131-148. 

The principal goal of turbulence research is to place turbulent flows on as sound a footing as we now have for laminar flows. For laminar flows we can start with Newton s laws of motion (generally in the form of the Navier-Stokes equations in Sec. 7.9), and from a description of the flow boundaries and the... 

Axially symmetric nonswirling turbulent flow of continuous incompressible Newton s fluid was considered. In this case the generalized equation of substance (mass, impulse, heat, turbulence kinetic energy) transfer is ... 

Differential equations (3.33) and (3.34) are easily solved for stationary regime of axial symmetric non-swirling turbulent flow of incompressible Newton two-phase medium without consideration of inter-phase heat- and mass-transfer. That is why the force of inter-phase interaction... 

An integral form of the general momentum balance will be obtained by applying Newton s second law of motion to a control volume of fluid. The evaluation of the resulting integral necessitates a knowledge of the velocity profile and appropriate assumptions of its form must be made for both laminar and turbulent flow conditions. 

In turbulent flow, momentum transfer is often characterized by an eddy viscosity Ey defined by analogy with Newton s Law of Viscosity. The time averaged momentum flux (shear stress) in the y direction owing to the gradient of the time averaged velocity in Ak z direction is given by... 

The first type of flow at low velocities where the layers of fluid seem to slide by one another without eddies or swirls being present is called laminar flow and Newton s law of viscosity holds, as discussed in Section 2.4A. The second type of flow at higher velocities where eddies are present giving the fluid a fluctuating nature is called turbulent flow. 

For nonisothermal systems a general differential equation of conservation of energy will be considered in Chapter 5. Also in Chapter 7 a general differential equation of continuity for a binary mixture will be derived. The differential-momentum-balance equation to be derived is based on Newton s second law and allows us to determine the way velocity varies with position and time and the pressure drop in laminar flow. The equation of momentum balance can be used for turbulent flow with certain modifications. 

Equation 3-13 is often called Newton s law. In the regime of Newton s law. the drag coefficient of a sphere is approximately 0.44, as shown in Figure 3-2, Newton s law applies to turbulent flow regimes. 

Newton s law is formulated so that the rate of shear (duy/dz) plays the role of a driving force while the force per unit area plays the role of a rate variable. This seems like a role reversal, but defining Newton s law in this way corresponds to a viscosity coefficient that is larger for more viscous fluids. Newton s law is valid only for laminar flow, which means flow in layers. Flow that is not laminar is called turbulent flow, and Newton s law does not hold for turbulent flow. There are some liquids, such as blood and polymer solutions, that do not obey Newton s law even for laminar flow. These fluids are called non-Newtonian fluids or thixotropic fluids and can be described by a viscosity coefficient that depends on the rate of shear. 

For Reynolds numbers > 1000, the flow is fully turbulent. Inertial forces prevail and becomes constant and equal to 0.44, the Newton region. The region in between Re = 0.2 and 1000 is known as the transition region andC is either described in a graph or by one or more empirical equations. 

The situation with regard to convective (turbulent) momentum transport is somewhat more complex because of the tensor (dyadic) character of momentum flux. As we have seen, Newton s second law provides a correspondence between a force in the x direction, Fx, and the rate of transport of x-momentum. For continuous steady flow in the x direction at a bulk... 

When a fluid flows past a solid surface, the velocity of the fluid in contact with the wall is zero, as must be the case if the fluid is to be treated as a continuum. If the velocity at the solid boundary were not zero, the velocity gradient there would be infinite and by Newton s law of viscosity, equation 1.44, the shear stress would have to be infinite. If a turbulent stream of fluid flows past an isolated surface, such as an aircraft wing in a large wind tunnel, the velocity of the fluid is zero at the surface but rises with increasing distance from the surface and eventually approaches the velocity of the bulk of the stream. It is found that almost all the change in velocity occurs in a very thin layer of fluid adjacent to the solid surface ... 

In a supersonic gas flow, the convective heat transfer coefficient is not only a function of the Reynolds and Prandtl numbers, but also depends on the droplet surface temperature and the Mach number (compressibility of gas). 154 156 However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature defined in Ref. 623. Thus, the convective heat transfer coefficient may still be estimated using the experimental correlation proposed by Ranz and Marshall 505 with appropriate modifications to account for various effects such as turbulence,[587] droplet oscillation and distortion,[5851 and droplet vaporization and mass transfer. 555 It has been demonstrated 1561 that using the modified Newton s law of cooling and evaluating the heat transfer coefficient at the film temperature allow numerical calculations of droplet cooling and solidification histories in both subsonic and supersonic gas flows in the spray. 

The idealised flow of non-reactive liquids in smooth channels has been considered by founding physicists such as Newton and Poiseuille and is well understood. Slow flow of a liquid is laminar or streamlined while rapid flow is turbulent with the transition occurring when the Reynolds number - the dimensionless parameter (Udphf) in which U is the velocity of the liquid, p and i] are the density and... 

The flow regime in a vessel under mixing may be laminar or turbulent. Under laminar conditions, may be expressed in terms of the stress obtained from Newton s... 

A. N. Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction effect of the variation of rheological parameters. J. Non-Newton. Fluid 1998, 79 (2-3), 433-468. 

Weber [572] has recently drawn attention to this problem by reminding us of the fact that fully developed turbulence cannot be realized in unbaffied laboratory vessels. Only at Re > 10 does the friction factor Cf in stirrer flow (counterpart of the Newton number) become constant. This can only be attained by stirring water in tanks with D > 1 m ... 

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