Time averaging, Reynolds

Classical models Based on (time-averaged) Reynolds equations. 

Following Reynolds, a time-averaged quantity ( ) I is defined ... 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

When the continmty equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = X, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side ... 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu, -pv, and -pw are always non-zero beeause they eontain squared veloeity fluetuations. The shear stresses -pu v, -pu w, -pv w and are assoeiated with eorrelations between different veloeity eomponents. If, for instanee, u and v were statistieally independent fluetuations, the time average of their produet u v would be zero. However, the turbulent stresses are also non-zero and are usually large eompared to the viseous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

Hydrodynamic effects on suspended particles in an STR may be broadly categorized as time-averaged, time-dependent and collision-related. Time-averaged shear rates are most commonly considered. Maximum shear rates, and accordingly maximum stresses, are assumed to occur in the impeller region. Time-dependent effects, on the other hand, are attributable to turbulent velocity fluctuations. The relevant turbulent Reynolds stresses are frequently evaluated in terms of the characteristic size and velocity of the turbulent eddies and are generally found to predominate over viscous effects. 

Reynolds stress, time average of the product of the velocity deviations in the axial and radial direction volume, ft3 velocity, ft/hr terminal velocity, ft/hr velocity in the normal direction, ft/hr specific volume, ft3/lb... 

The focus of RANS simulations is on the time-averaged flow behavior of turbulent flows. Yet, all turbulent eddies do contribute to redistributing momentum within the flow domain and by doing so make up the inherently transient character of a turbulent-flow field. In RANS, these effects of the full range of eddies are made visible via the so-called Reynolds decomposition of the NS equations (see, e.g., Tennekes and Lumley, 1972, or Rodi, 1984) of the flow variables into mean and fluctuating components. To this end, a clear distinction is required between the temporal and spatial scales of the mean flow on the one hand and those associated with the turbulent fluctuations on the other hand. 

It follows that if an element of fluid moves in they-direction in a region where the mean velocity gradient dvjdy is zero, a fluctuation v y gives rise, on average, to a zero fluctuation v x. The time-average product of the fluctuations (the Reynolds stress) is zero and the fluctuations are said to be uncorrelated. 

In the description of turbulent fluctuations it has been useful to employ the nomenclature and approach developed by Reynolds (R2). In this instance the instantaneous velocity is made up of two terms, the time-average velocity and the fluctuating velocity as indicated in the following expressions ... 

It should be recognized that the boundary conditions of the problem will establish the value of the hydrodynamic velocity, u. In the case of most turbulent flows the indirect influence of molecular diffusion on the hydro-dynamic velocity can be neglected. It should be emphasized that the hydrodynamic velocity is the time-average point velocity in Reynolds sense (R2). Under unsteady, nonuniform conditions of flow between parallel plates the material balance may be expressed for turbulent flow in the following form ... 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-empirical models to express the Reynolds stresses in terms of time-averaged velocities. This is the closure problem of turbulence. In all but the simplest geometries, numerical methods are required. 

The most widely adopted method for the turbulent flow analysis is based on time-averaged equations using the Reynolds decomposition concept. In the following, we discuss the Reynolds decomposition and time-averaging method. There are other methods such as direct numerical simulation (DNS), large-eddy simulation (LES), and discrete-vortex simulation (DVS) that are being developed and are not included here. 

In using the Reynolds decomposition, closure of the time-averaged Navier-Stokes equations cannot readily be realized because of the unknown correlation terms such as turbulent... 

Reynolds stresses generated by time averaging. Thus, additional equations are needed to correlate these terms with time-averaged quantities. These additional equations may come from turbulence models. The two most commonly used turbulence models, the mixing length model and the k-e model, are introduced. 

It is assumed that the instantaneous Navier-Stokes equations for turbulent flows have the exact form of those for laminar flows. From the Reynolds decomposition, any instantaneous variable, (j>, can be divided into a time-averaged quantity and a fluctuating part as... 

The purpose of the time averaging after the volume averaging is to express averages of products in terms of products of averages and to account for turbulent fluctuations and high-frequency fluctuations [Soo, 1989]. The volume-time averaging is presented here in a similar way to that of the Reynolds analysis of single-phase turbulent flow. 

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