Turbulent flow time-averaging

Under the conditions of turbulence, the time-averaged velocity field is symmetric with respect to the free stagnation plane, provided the flow rates from the two nozzles are equal. The mean axial velocity profile has a similar shape to the curve of uju ) vs x. The gradient of the time-averaged axial velocity takes the maximum at the stagnation plane, while it approaches zero near the nozzle. 

Similarity coordinates and 77 were used to sample the flow fleld over many turbulent cycles. Time averaging in these coordinates was used to find both the mean and rms profiles of the wall jet and boundary layer flow without resorting to turbulence modeling. 

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part ... 

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). 

As may be expected, turbulent flow (9,11) is more efficient for droplet formation in low viscosity Hquids. With the average amount of energy dissipated per unit time and volume equal to Z and mass density equal to p, the larger eddies are characterized by a velocity gradient equal to... 

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-... 

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. 

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. 

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 ... 

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. 

A record of the axial velocity component vx for steady turbulent flow in a pipe would look like the trace shown in Figure 1.22. The trace exhibits rapid fluctuations about the mean value, which is determined by averaging the instantaneous velocity over a sufficiently long period of time. Figure 1.22 shows the case in which the mean velocity remains constant this is therefore known as steady turbulent flow. In unsteady turbulent flow, the mean value changes with time but it is still possible to define a mean value because, in practice, the mean will drift slowly compared with the frequency of the fluctuations. 

---