Turbulence time averaging

Nonintrusive Instrumentation. Essential to quantitatively enlarging fundamental descriptions of flow patterns and flow regimes are localized nonintmsive measurements. Early investigators used time-averaged pressure traverses for holdups, and pilot tubes for velocity measurements. In the 1990s investigators use laser-Doppler and hot film anemometers, conductivity probes, and optical fibers to capture time-averaged turbulent fluctuations (39). 

To analy2e premixed turbulent flames theoretically, two processes should be considered (/) the effects of combustion on the turbulence, and (2) the effects of turbulence on the average chemical reaction rates. In a turbulent flame, the peak time-averaged reaction rate can be orders of magnitude smaller than the corresponding rates in a laminar flame. The reason for this is the existence of turbulence-induced fluctuations in composition, temperature, density, and heat release rate within the flame, which are caused by large eddy stmctures and wrinkled laminar flame fronts. 

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. 

Time Averaging In turbulent flows it is useful to define time-averaged and fluctuation values of flow variables such as velocity com-... 

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-... 

TURBULENCE ON TIME-AVERAGED NAVIER-STOKES EQUATIONS ... 

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. 

For most applications, the engineer must instead resort to turbulence models along with time-averaged Navier-Stokes equations. Unformnately, most available turbulence models obscure physical phenomena that are present, such as eddies and high-vorticity regions. In some cases, this deficiency may partially offset the inherent attractiveness of CFD noted earlier. 

When using LES, the time-dependent three-dimensional momentum and continuity are solved for. A subgrid turbulence model is used to mode the turbulent scales that are smaller than the cells. Instead of the traditional time averaging, the equations for using LES are filtered in space, and is a function of space and time. 

With LES we get much more information than with traditional time-averaged turbulence models, since we are resolving most of the turbulence. In Fig. T1.15 the computed u velocity is shown as a function of time in two cells one cell is located in the wall jet (Fig.. 15a), and the other cell is in the middle of the room (Fig. ll.lSh). It is found the instantaneous fluctuations are very large. For example, in the region of the wall jet below the ceiling where the time-averaged velocity u)/l] ) is typically 0.5, the instantaneous velocity fluctuations are between 0.2 and 0.9. In the middle of the room, which is a low-velocity region, the variation of u is much slower, i.e., the frequency is lower. 

In the present discussion only the problem of steady flow will be considered in which the time average velocity in the main stream direction X is constant and equal to ux. in laminar flow, the instantaneous velocity at any point then has a steady value of ux and does not fluctuate. In turbulent flow the instantaneous velocity at a point will vary about the mean value of ux. It is convenient to consider the components of the eddy velocities in two directions—one along the main stream direction X and the other at right angles to the stream flow Y. Since the net flow in the X-direction is steady, the instantaneous velocity w, may be imagined as being made up of a steady velocity ux and a fluctuating velocity ut, . so that ... 

In turbulent flow there is a complex interconnected series of circulating or eddy currents in the fluid, generally increasing in scale and intensity with increase of distance from any boundary surface. If, for steady-state turbulent flow, the velocity is measured at any fixed point in the fluid, both its magnitude and direction will be found to vary in a random manner with time. This is because a random velocity component, attributable to the circulation of the fluid in the eddies, is superimposed on the steady state mean velocity. No net motion arises from the eddies and therefore their time average in any direction must be zero. The instantaneous magnitude and direction of velocity at any point is therefore the vector sum of the steady and fluctuating components. 

In most situations a fluid would be turbulent implying that the velocity vector, as well as the concentration c exhibits considerable variability on time scales smaller than those of prime interest. This situation can be described by writing these quantities as the sum of an average quantity (normally a time average) and a perturbation... 

Turbulent flow reactors are modeled quite differently from laminar flow reactors. In a turbulent flow field, nonzero velocity components exist in all three coordinate directions, and they fluctuate with time. Statistical methods must be used to obtain time average values for the various components and to characterize the instantaneous fluctuations about these averages. We divide the velocity into time average and fluctuating parts ... 

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. 

K. L. McCarthy, L. Odberg, R. L. Powell 1994, (Turbulent pipe-flow studied by time-averaged NMR imaging - measurements of velocity profile and turbulent intensity), Magn. Reson. Imag. 12, 923. 

The velocity field in turbulent flow can be described by a local mean (or time-average) velocity, upon which is superimposed a time-dependent fluctuating component or eddy. Even in one-dimensional flow, in which the overall average velocity has only one directional component (as illustrated in Fig. 6-3), the turbulent eddies have a three-dimensional structure. Thus, for the flow illustrated in Fig. 6-3, the local velocity components are... 

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. 

The research on the flow regimes in packed tubes suggests that laminar flow CFD simulations should be reasonable for Re <100 approximately, and turbulent simulations for Re >600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real time-dependent turbulent flow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 time-averaged picture of the flow field. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. 

For laminar conditions of slow flow, as in candle flames, the heat transfer between a fluid and a surface is predominately conductive. In general, conduction always prevails, but in the unsteadiness of turbulent flow, the time-averaged conductive heat flux between a fluid and a stationary surface is called convection. Convection depends on the flow field that is responsible for the fluid temperature gradient near the surface. This dependence is contained in the convection heat transfer coefficient hc defined by... 

In turbulent flow, properties such as the pressure and velocity fluctuate rapidly at each location, as do the temperature and solute concentration in flows with heat and mass transfer. By tracking patches of dye distributed across the diameter of the tube, it is possible to demonstrate that the liquid s velocity (the time-averaged value in the case of turbulent flow) varies across the diameter of the tube. In both laminar and turbulent flow the velocity is zero at the wall and has a maximum value at the centre-line. For laminar flow the velocity profile is a parabola but for turbulent flow the profile is much flatter over most of the diameter. 

It is important to note that in deriving the shear stress distribution no assumption was made as to whether the fluid was Newtonian or whether the flow was laminar. In the case of turbulent flow, it is the time-averaged values of rrx and tw that are given by equations 1.40 and 1.41. In Section 1.13 these time-averaged stresses will be denoted by frx and tw. 

In general, the time-averaged value of the product of the fluctuations is non-zero so there is an additional flux of x-momentum in the y-direction due to the velocity fluctuations v x and v y. This momentum flux is equivalent to an extra apparent shear stress acting in the x-direction on the plane normal to the y-coordinate direction. Consequently, the mean total shear stress for turbulent flow can be written as... 

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