# Time average, turbulence

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. [c.1049]

A more quantitative description can be obtained by dividing the field conceptually into three regions a turbulent core in which momentum, heat, and mass are readily transported radially by strong eddying motions a thin, nearly laminar, sublayer hugging the wall and a somewhat thicker transition or buffer layer bridging the two. Within the laminar sublayer, viscous stresses are important and the steady-state Navier-Stokes equations are valid. In the other regions they are inadequate. The fluctuating velocities in these regions can be described by the time-dependent Navier-Stokes equations, but the solution is enormously complex. It has been estimated that modem computers would require a time equal to the age of the universe to adequately describe turbulent pipe flow at a Reynolds number of 10. To avoid this complexity, the time-dependent equations can be averaged in the manner originally suggested by Reynolds. This yields equations that are similar to those for laminar flow, but which contain additional terms involving time-averaged [c.89]

Each of the three time-averaged momentum equations contains three unknown turbulent stresses, pu[u j, commonly termed Reynolds stresses, only six of which are independent. The Reynolds stress pu[ for example, is the rate at which x-momentum, pu[, is being transported in the jy-direction by the velocity fluctuation u 2. A hierarchy of equations for velocity correlation functions, ie, averages of products of ti-, can be obtained, but each equation so derived involves an unknown higher order correlation function and hence the set of equations is not closed. A turbulence model is needed to determine the turbulent transport terms before the set of equations can be solved. Turbulence modeling is concerned with the development and testing of closure assumptions for the Reynolds stresses. A large number of closure models are available. They are usually divided into two groups, eddy viscosity models and Reynolds stress models, according to whether or not the Boussinesq assumption is appHed. [c.102]

Because u can be positive or negative, its impact on mixing is best quantified by calculating the root mean square (RMS) fluctuation. The RMS velocity is obtained by squaring the fluctuation, time averaging, and then taking the square root. Turbulent intensity is defined as ratio of RMS velocity to average velocity. Typically, turbulent intensity ranges from 0.5—0.7 near the impeller tip, to 0.05—0.15 in other parts of the tank. [c.423]

Time Averaging In turbulent flows it is useful to define time-averaged and fluctuation values of flow variables such as velocity com- [c.671]

The actual and fluctuating velocity components are, in general, functions of the three spatial coordinates x, y, and z and of time t. The time-averaged velocity is independent of time for a stationary flow. Nonstationary processes may oe considered where averages are defined over time scales long compared to the time scale of the turbulent fluctuations, but short compared to longer time scales over which the time-averaged flow variables change due, for example, to time-varying boundaiy conditions. The time average over a time interval 2T centered at time t of a turbulently fluctuating variable (t) is defined as [c.671]

Sea salt particles are the biggest contributor by mass of particulate material into the marine atmosphere, with 10 -10 ° tonnes cycled through the atmosphere annually. They tend to be relatively large, typically 0.1-1 /an in diameter. This large size means that there is a significant fall-out of particles within the marine boundary layer (up to 90%). Fiowever, those that are carried by turbulence into the free atmosphere in concentrations of typically several /rg m are larger than average CCN and so play a dominant role in the important coalescence mode of rain formation. This role is enhanced by the strongly hygroscopic nature of the largely NaCl sea salt aerosol. A relative humidity of only 75% is required for the initiation of condensation around a NaCl nucleus. While some salts have even lower thresholds—KjCOj at 44%, for instance—the abundance of atmospheric sea salt makes this a significant source of cloud droplets. [c.25]

TURBULENCE ON TIME-AVERAGED NAVIER-STOKES EQUATIONS [c.792]

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. [c.794]

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. [c.1048]

The turbulent fluidized bed has a similar or slightly lower soHds volume fraction than the vigorously bubbling bed. There is considerable transport of soHds out of the turbulent bed and the bed level is not very distinct. Large-scale cyclones are needed to return soHds to the bed. On average, the bed inventory passes through the cyclones several times per hour. [c.74]

Control of bubble size via particle-size control is easy using Group A powders. A typical Group A powder, having an average particle size of 70 p.m and a normal particle size distribution, has nearly 25% of the particles as fines, ie, below 44 p.m. If these particles are spherical and if iaterparticle forces are absent, bubble size is typically smaller than 25 mm (see Fig. 5a). Well-defined bubbles are not present in the typical industrial turbulent fluidized-bed reactor. The turbulent fluid bed can be viewed as a single-phase system for most practical design considerations. Bubble size control is more difficult for Group B powders, especially in appHcations where there is Htfle control of particle properties, eg, in processing of naturally occurring materials. In some cases, increasing the gas velocity and moving into turbulent or fast-fluidization regimes has been the preferred solution. [c.75]

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 [c.101]

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows [c.108]

In turbulent mixing, the Hquid velocity at any point u can be considered the sum of an average velocity ul and a fluctuating (with time) velocity u [c.423]

Plasma diagnostics, the deterrnination of conditions within plasmas, also refers to the broad collection of experimental techniques and associated cahbration and analytical methods used to assess the characteristics of plasmas. Noteworthy properties include the identities, concentrations, and energy distributions of the various particle species such as neutrals, electrons, and ions, and thek velocity distributions as functions of space and time. Quantities such as plasma flow velocities, turbulence, instabiUties, and flow of energy into and out of plasmas by various means also are desked. Most diagnostic methods have limited resolution spatially, temporally, and spectrally. Therefore, plasma characteristics that are derived from measurements generally are averaged. Measured quantities in plasma diagnostics usually are integrated along a line of sight through the plasma to the instmmentation, yielding spatially integrated results. It is possible, however, to measure point temperatures within low temperature plasmas using fluoroptic thermometers (30). No one method for assaying plasmas is universally appHcable. A variety of diagnostic tools is needed to characterize a plasma empirically and to compare its empirical and theoretical characteristics. [c.111]

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 [c.197]

For turbulent flow through shallow tube banks, the average friction factor per row will be somewhat greater than indicated by Figs. 6-42 and 6-43, which are based on 10 or more rows depth. A 30 percent increase per row for 2 rows, 15 percent per row lor 3 rows and 7 percent per row for 4 rows can be taken as the maximum likely to be encountered (Boucher and Lapple, Chem. Eng. Prog., 44, 117—134 [1948]). [c.663]

While the above equation gives the relationship between pressure and flow from a macroscopic point of view, it does not explain what is going on inside the valve. Valves create a resistance to flow by restricting the cross sectional area of the flow passage and also by forcing the fluid to change direction as it passes through the body and trim. The consei vation of mass principle dictates that, for steady flow, density X average velocity X cross sec tional area equals a constant. The average velocity of the fluid stream at the minimum restriction in the valve is therefore much higher than at the inlet. Note that due to the abrupt nature of the flow contraction that forms the minimum passage, the main fluid stream may separate from the passage walls and form a jet that has an even smaller cross section, the so-called vena contrac ta. The ratio of minimum stream area to the corresponding passage area is called the contraction coefficient. As the fluid expands from the minimum cross sectional area to the full passage area in the downstream piping, large amounts of turbulence are generated. Direction changes can so induce significant amounts of turbulence. [c.787]

The difficulty with Eq, (26-58) is that it is impossible to determine the velocity at every point, since an adequate turbulence model does not currently exist, The solution is to rewrite the concentration and velocity in terms of an average and stochastic quantity C = (C) -t- C Uj = (uj) + Uj, where the brackets denote the average value and the prime denotes the stochastic, or deviation variable. It is also helpful to define an eddy diffusivity Kj (with units of area/time) as [c.2342]

Near-surface (within 10 m of the ground) meteorological instrumentation always includes wind measurements and should include turbulence measurements as well. Such measurements can be made at 10 m above ground by using a guyed tower. A cup anemometer and wind vane (Fig. 19- 7), or a vane with a propeller speed sensor mounted in front (Fig. 19-8), can be the basic wind system. The wind sensor should have a threshold starting speed of less than 0.5 m s an accuracy of 0.2 m s or 5%, and a distance constant of less than 5 m for proper response. The primary quantity needed is the hourly average wind speed. A representative value may be obtained from values taken each minute, although values taken at intervals of 1-5 sec are better. [c.306]

The power dissipation correlations discussed are specifically applicable to the mixing of fluids that are Newtonian. A Newtonian fluid has a viscosity independent of the shear to which the fluid is subjected. Thus, although the shear rates vary greatly throughout the fluid in an agitated vessel, the viscosity of a Newtonian fluid will be the same at all points in the vessel. In contrast, the apparent viscosity of a non-Newtonian fluid at any point in the vessel depends on the magnitude of either the shear stress or the shear rate at that point and may also depend upon the previous history of the fluid. If the apparent viscosity decreases with increasing shear rate, the fluid is called pseudoplastic. If the apparent viscosity increases with shear rate, the fluid is termed dilatant If the apparent viscosity of a pseudoplastic fluid decreases with the time a particular rate of shear is applied, the fluid is termed thixotropic. The Newtonian/non-Newtonian behavior of a fluid or fluid mixture is known only from prior experience or by viscometry. Many non-Newtonian fluids encountered are pseudoplastic. For these fluids studies have established that the power number versus impeller Reynolds number behavior is such that the power number is always equal to or less than the value for a Newtonian fluid. Thus, for pseudoplastic fluids. Figures 20 and 22 can still be used and will provide conservative numbers. If the agitation conditions are known to be turbulent and result in operation on the flat portion of the power number-Reynolds number curve, a precise knowledge of apparent viscosity is not needed. However, in the laminar range, a knowledge of average shear and thus apparent viscosity is needed to calculate the appropriate Reynolds number. In a stirred tank the average shear rate is equal to approximately 10 times the rotational speed of the impeller. Experimental measurements are required to determine apparent viscosities at this shear rate. [c.463]

DEGADIS (Dense GAs Dispersion) models atmospheric dispersion from ground-le el, area sources heavier-than air or neutrally buoyant vapor releases with negligible momentum or as a jet from pressure relief into the atmosphere with flat unobstructed terrain. It simulates continuous, instantaneous or finite-duration, time-invariant gas and aerosol releases, dispersion with gravity-driven flow contaminant entrainment into the atmosphere and downwind travel. It accounts for reflection of the plume at the ground, and has options for isothermal or adiabatic heal transfer between the vapor cloud and the air. Variable concentration averaging times may be used and time-varying source-term release rate and temperature profiles may be used. It has been validated with large-scale field data. It models induced ground turbulence from surface roughness. [c.351]

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). [c.514]

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. [c.497]

Turbulent Transport and Diffusion. There are two pollutant transport terms in equation 5 an advection term, in which pollutants are carried along with the time-averaged mean wind flow and a dispersion term representing transport resulting from local turbulence. The averaging time that deterrnines the mean winds is related to the spatial scale of the system being modeled. Minutes may be appropriate for urban-scale simulations, multihour averages for the regional scale, and daily to weekly averages for determining long-term concentrations of nonreactive pollutants. [c.381]

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. [c.518]

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. [c.632]

Stirred Tank Agitation Turbine impeller agitators, of a variety of shapes, are used for stirred tanks, predominantly in turbulent flow. Figure 6-39 shows typical stirred tank configurations and time-averaged flow patterns for axial flow and radim flow impellers. In order to prevent formation of a vortex, four vertical baffles are normally installed. These cause top-to-bottom mixing and prevent mixing-ineffective swirhng motion. [c.660]

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- [c.671]

An additional turbulence pressure term equal to —V Jcbij, where k = turbulent kinetic energy and 1 i=J and 6, = 0 if i 9 J, is sometimes included in the right-hand side. To solve the equations of motion using the Boussinesq approximation, it is necessary to provide equations for the single scalar unknown [Lt (and /c, if used) rather than the nine unknown tensor components With this approximation, and using the effective viscosity PefF = M- + h, ihe time-averaged momentum equation is similar to the original Navier-Stokes equation, with time-averaged variables and [L f replacing the instantaneous variables and molecular viscosity. However, solutions to the time-averaged equations for turbulent flow are not identical to those for laminar flow because PefF is not a constant. [c.672]

CFD methods are used for incompressible and compressible, creeping, laminar and turbulent, Newtonian and non-Newtonian, and isothermal and nonisothermal flows. Chemically reacting flows, particularly in the field of combustion, have been simulated. Solution accuracy must be considered from several perspectives. These include convergence of the algorithms for solving the nonlinear discretized equations and convergence with respect to refinement of the mesh so that the discretized equations better approximate the exact equations and, in some cases, so that the mesh more accurately fits the true geometry. The possibility that steady-state solutions are unstable must ways be considered. In addition to numerical sources of error, modeling errors are introduced in turbulent flow, where semiempirical closure models are used to solve time-averaged equations of motion, as discussed previously. Most commercial CFD codes include the /c— turbulence model, which has been by far the most widely used. More accurate models, such as differential Reynolds stress and renormahza-tion group theory models, are also becoming available. Significant solution error is known to result in some problems from inadequacy of the turbulence model. Closure models for nonhnear chemical reaction source terms may also contribute to inaccuracy. Direct numerical simulation and large eddy simulation, which involve solutions for velocity fluctuations, under limited conditions or with certain modehng assumptions, remain primarily research areas. [c.673]

Figure 6-56 gives an example CFD calculation for time-dependent flow past a square cylinder at a Reynolds number of 22,000 (Choud-hury, et al., Trans. ASME Fluids Div., Lake Tahoe, Nev. [1994]). The computation was done with an implementation of the renormalization group theory /c— model. The series of contour plots of stream function shows a sequence in time over about 1 vortex-shedding period. The calculated Stroiibal number (Eq. [6-195]) is 0.146, in excellent agreement with experiment, as is the time-averaged drag coefficient, Co = 2.24. Similar computations for a circular cylinder at Re = 14,500 have given excellent agreement with experimental measurements for St and Co (Introduction to the Renormalization Group Method and Turbulence Modeling, Fluent, Inc., 1993). [c.674]

The principal framework of empirical equations which form a basis for estimahng concentrations from point sources is commonly referred to as the Gaussian plume model. Employing a three-dimensional axis system of downwind, crosswind, and vertical with the origin at the ground, it assumes that concentrations from a continuously emitting plume are proportional to the emission rate, that these concentrations are diluted by the wind at the point of emission at a rate inversely proportional to the wind speed, and that the time- averaged (about 1 h) pollutant concentrations crosswind and vertically near the source are well described by Gaussian or normal (bell-shaped) distributions. The standard deviations of plume concentration in these two directions are empirically related to the levels of turbulence in the atmosphere and increase with distance from the source. [c.296]

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. [c.825]

A distribution of veloeities in a fluid gives rise to a transport of momentum in the fluid in eomplete analogy with the transport of energy whieh results from a distribution of temperatures. To analyse this transport of momentum in a fluid with a gradient in the average loeal veloeity, we use the same method as employed in the ease of themial eonduetion. That is, we eonsider a layer of fluid eontained between two parallel planes, niovhig with veloeities in the v-direetion with values Uj and t/, U2> Uj, as illustrated in figure A3.1.5. We suppose that the width of the layer is very large eompared with a mean free path, and that the fluid adjaeent to the moving planes moves with the veloeity of the adjaeent plane. If the veloeities are not so large as to develop a turbulent flow, then a steady state ean be maintained with an average loeal veloeity, u(x,y,z), in the fluid of the fomi, u(x,y,z) = (z)x, where xis a unit veetor m the v-direetion. [c.673]

Turbulence in the dow of a premixture dattens the velocity profile and increases the effective burning velocity of the mixture eg, at a pipe-Reynolds number of 40,000 the turbulent burning velocity is several times the laminar burning velocity and it can be perhaps fifty times larger at very high Reynolds numbers. A turbulent dame is always somewhat noisy, the apparent dame surface becomes diffuse owing to the ductuations in the actual or dame surface about its average position, and its stabiHty tends to be less predictable. The instantaneous dame surface may be thought of as wrinkled by velocity variations in turbulent dow, or by the average distribution over a greater thickness (or time). Although the resulting enhancement of the mixture consumption rate may be considerable, turbulence is often considered undesirable in Bunsen-type dames. For this and other reasons, a large number of burner ports of small characteristic dimension, rather than a single large port, are frequendy used to assure laminar dow to the individual dames. However, turbulence has an essential role in faciHtating the mixing of fuel, oxidizer, and dame products, and serves an important function in the various types of dame-stabilizers of practical importance. [c.524]

Pasteurization. All egg products must be pasteurized to render them Salmonella-n.e iXbj e. Conventional plate-type pasteurizers having the usual attachments, including holding tubes, flow diversion valve, regeneration cycle, etc, are used (see Sterilization). Minimum pasteurization requirements are based on the bacterial kill obtained when beating whole egg to 60°C for a holding time of 3.5 min. Because of viscosity, flow of egg Hquid is laminar, as opposed to turbulent, through the holding tubes. Because flow is laminar, the holding time of the fastest particle is only one-half that of the average holding time the average holding time is 3.5 min, but holding time is actually 1.75 min for the fastest particles. The ultrapasteurization flow through the holding tubes is turbulent flow because of high velocities used. [c.459]

The averaging time of the rapid-response record [Fig. 4-1 (a)] is an inherent characteristic of the instrument and the data acquisition system. It can become almost an instantaneous record of concentration at the receptor. However, in most cases this is not desirable, because such an instantaneous record cannot be put to any practical air pollution control use. What such a record reveals is something of the turbulent structure of the atmosphere, and thus it has some utility in meteorological research. In communications [c.42]

The average density of turbulent sites x > 1), for example, is the same whether the averaging is performed over space, time or both. While the transition region also exhibits scaling properties characteristic of continuous phase transitions,+ and it has been conjectured that the system is in the same universality class as directed percolation [Pomeau86], it does not appear to be universal [chate90]. The critical exponents characterizing the distribution of laminar clusters at the transition threshold for C, = l,for example, are different for s = 2.1 and s = 3 and do not agree with the corresponding value for directed percolation. Indeed, the two CA approximation schemes about to be outlined below were both motivated, in part, by this observed inequality [chate90]. [c.402]

These properties include (l)a continuous decrease in the average density of turbulent sites as the threshold is approached from above (with exponent /3) (2) divergence of the average transient time (3) algebraic distributions of the sizes and durations of laminar clusters at the threshold (with exponents and < ). Houlrik, Webrnan arid Jensen [houl90] discuss the critical behavior of this map from a mean-field theoretic perspective. [c.402]

See pages that mention the term

**Time average, turbulence**:

**[c.672] [c.46] [c.100] [c.638] [c.657] [c.300] [c.353]**

Modeling of chemical kinetics and reactor design (2001) -- [ c.792 , c.793 ]