Turbulent boundary layer model

CATALYSTS - REGENERATION - FLUID CATALYTIC CRAC KING UNITS] (Vol 5) Turbulent boundary layer model... 

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

The next step is to introduce the limiting effect of convection into the problem. A solution based on a nonlinear turbulent boundary layer model has been given by Foster It states that the critical time C at the onset of convection is given as... 

Four of the simplest and best known of the theories of mass transfer from flowing streams are (1) the stagnant-film model, (2) the penetration model, (3) the surface-renewal model, and (4) the turbulent boundary-layer model... 

The turbulent boundary layer model accounts for the transfer of a solute molecule A from a turbulent stream to a fixed surface. Eddy diffusion is rapid in the turbulent stream and molecular diffusion is relatively insignificant. It is supposed that the turbulence is damped out in the immediate vicinity of the surface. In the intermediate neighborhood between the turbulent stream and the fixed surface, it is supposed that transport is by both molecular and eddy diffusion which take place in parallel. The total rate of transfer (moles of A transferred per unit time per unit area) is given by an extended form of Fick s law... 

The approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

Any consideration of mass transfer to or from drops must eventually refer to conditions in the layers (usually thin) of each phase adjacent to the interface. These boundary layers are envisioned as extending away from the interface to a location such that the velocity gradient normal to the general flow direction is substantially zero. In the model shown in Fig. 8, the continuous-phase equatorial boundary layer extends to infinity, but the drop-phase layer stops at the stagnation ring. At drop velocities well above the creeping flow region there is a thin laminar sublayer adjacent to the interface and a thicker turbulent boundary layer between this and the main body of the continuous phase. 

A more realistic description that avoids the countercurrent flow in the region between two successive paths is represented in Fig. 6b, but we will assume that the regions 2 have a much smaller extent than the regions 1. It is appropriate to cite here references [59,60] in which the Danckwerts renewal idea was used to describe the turbulent boundary layer near a wall, as well as the 1969 paper of Black [61] in which a model similar to that of Ruckenstein [58] is considered. 

In the seventies, the growing interest in global geochemical cycles and in the fate of man-made pollutants in the environment triggered numerous studies of air-water exchange in natural systems, especially between the ocean and the atmosphere. In micrometeorology the study of heat and momentum transfer at water surfaces led to the development of detailed models of the structure of turbulence and momentum transfer close to the interface. The best-known outcome of these efforts, Deacon s (1977) boundary layer model, is similar to Whitman s film model. Yet, Deacon replaced the step-like drop in diffusivity (see Fig. 19.8a) by a continuous profile as shown in Fig. 19.8 b. As a result the transfer velocity loses the simple form of Eq. 19-4. Since the turbulence structure close to the interface also depends on the viscosity of the fluid, the model becomes more complex but also more powerful (see below). 

The principle of the model is to scan the bed surface, which is subdivided into boxes whose the width and the length are equal respectively to the spanwise and streamwise statistical periodicities of appearing of the coherent structures. In fact, some authors have shown that the phenomenon of ejection in a turbulent boundary layer could be connected with the particle s take-off. 

Taylor-Prandtl model for turbulent boundary layer flow. 

A numerical procedure for calculating the heat transfer rate with turbulent boundary layer flow was discussed in Chapter 5. This procedure used a mixing length-based turbulence model. Discuss the modifications that must be made to this procedure to apply it to mixed convective flow over a vertical plate. 

In 1968 a conference was held at Stanford on turbulent boundary layer prediction-method calibration (S3), where for the first time a large number of methods, totaling 29, were compared on a systematic basis. This comparison established the viability of prediction methods based on various closure models for the partial differential equations describing turbulent boundary layer flows, and has stimulated considerable more recent work on this approach. 

In boundary-layer calculations, most workers simply use zonal models, with Eq. (9) in the inner region [which becomes Eq. (8) further from the wall] and something like Eq. (7) in the outer portion of the flow. Byrne and Hatton (B5) use a three-layer model as the basis for vt assumptions. Mellor and Herring (M2) have used concepts from the theory of matched asjTnptotic expansions to obtain composite representation for I valid across an entire turbulent boundary layer. A t j"pical distribution of 1 in a boundary layer is shown in Fig. 5. 

The prediction of turbulent boundary-layer separation by MVF methods has not been very successful. Indeed, it may be appropriate to identify turbulent separation in terms of the turbulence near the wall, and this will require use of a more sophisticated model (i ITE or MRS), quite possibly in their full (rather than boundary-layer) form. 

Fully computational methods (FCM), using a variety of turbulence modelling methods, have been extensively applied to computing flows around single buildings in turbulent boundary layers. Using turbulence closure methods many models have predicted mean... 

Gaussian puff/plume dispersion modeling techniques embedded in D2PC are representative of the state of the art in the late 1970s. Since then, there have been many technical advances in understanding atmospheric turbulence, boundary layer structure, and the effects of complex terrain that could benefit the CSEPP program. 

Closing the k-e model by turbulence modeling we relate the unknown Reynolds stress tensor and the turbulent transport terms to the fundamental mean flow variables, or the scaled variables in turbulent boundary layers, introducing additional approximations. 

In this section the heat and mass transport coefficients for turbulent boundary layers are examined. In this case the model derivation is based on the governing Reynolds averaged equations. In these equations statistical covariances appear which involve fluctuating velocities, temperatures and concentrations. The nature of these terms is not known a priori and their effects must by estimated by semi-empirical turbulence modeling. The resulting parameterizations allow us to express the unknown turbulent fluctuations in terms of the mean flow field variables. It is emphasized that the Reynolds equations are not actually solved, merely semi-empirical relations are derived for the wall fluxes through the inner boundary layer. 

The modeling procedure can be sketched as follows. First an approximate description of the velocity distribution in the turbulent boundary layer is required. The universal velocity profile called the Law of the wall is normally used. The local shear stress in the boundary layer is expressed in terms of the shear stress at the wall. From this relation a dimensionless velocity profile is derived. Secondly, a similar strategy can be used for heat and species mass relating the local boundary layer fluxes to the corresponding wall fluxes. From these relations dimensionless profiles for temperature and species concentration are derived. At this point the concentration and temperature distributions are not known. Therefore, based on the similarity hypothesis we assume that the functional form of the dimensionless fluxes are similar, so the heat and species concentration fluxes can be expressed in terms of the momentum transport coefficients and velocity scales. Finally, a comparison of the resulting boundary layer fluxes with the definitions of the heat and mass transfer coefficients, indiates that parameterizations for the engineering transfer coefficients can be put up in terms of the appropriate dimensionless groups. 

The existing turbulence models consist of approximate relations for the /ij-parameter in (5.246). The Prandtl mixing-length model (1.356) represents an early algebraic (zero-equation) model for the turbulent viscosity Ht in turbulent boundary layers. 

Two traditional approaches to the closure of the Reynolds equation are outlined below. These approaches are based on Boussinesq s model of turbulent viscosity completed by Prandtl s or von Karman s hypotheses [276, 427]. For simplicity, we confine our consideration to the case of simple shear flow, where the transverse coordinate Y = Xi is measured from the wall (the results are also applicable to turbulent boundary layers). According to Boussinesq s model, the only nonzero component of the Reynolds turbulent shear stress tensor and the divergence of this tensor are defined as... 

In accordance with Boussinesq s model the turbulence boundary layer on a flat plate is described by the equations... 

Eddy Diffusivity Models. The mean velocity data described in the previous section provide the bases for evaluating the eddy diffusivity for momentum (eddy viscosity) in heat transfer analyses of turbulent boundary layers. These analyses also require values of the turbulent Prandtl number for use with the eddy viscosity to define the eddy diffusivity of heat. The turbulent Prandtl number is usually treated as a constant that is determined from comparisons of predicted results with experimental heat transfer data. 

The behavior of the alternate forms of eM/v in the near-wall region of a turbulent boundary layer is shown in Fig. 6.35. The classical Prandtl-Taylor model assumes a sudden change from laminar flow (eM/v = 0) to fully turbulent flow (Eq. 6.173) at y = 10.8. The von Kftrman model [88] allows for the buffer region and interposes Eq. 6.174 between these two regions. The continuous models depart from the fully laminar conditions of the sublayer around y+ = 5 and asymptotically approach limiting values represented by Eq. 6.173. In finite difference calculations, eM/v is allowed to increase until it reaches the value given by Eq. 6.158 and then is either kept constant at this value or diminished by an intermittency factor found experimentally by Klebanoff [92]. 

Uniform Fluid Properties. Analyses of turbulent boundary layers experiencing surface transpiration employ a hierarchy of increasingly complex models of the turbulent transport mechanisms. Most of the analyses, supported by complementary experiments, have emphasized the transpiration of air into low-speed airstreams [110-112], Under these conditions, the fluid properties in the boundary layer are essentially constant, and the turbulent boundary layer can be described mathematically with Eqs. 6.170 and 6.179. In addition, when small quantities of a foreign species are introduced into the boundary layer for diagnostic purposes or by evaporation, the local foreign species concentration in the absence of thermal diffusion is given by... 

Deacon [28] developed a boundary layer model based on turbulent fluid flow in the vicinity of a smooth rigid wall. By assuming that the wind stress is continuous across the air-water interface, producing a constant flux of momentum, the friction velocity on the water side can be determined as u = w a(9a/9w)° in which a and w refer to air and water, respectively. This approach has been found to provide a reasonable description of gas transfer in wind tunnels at low wind speeds [10]. Another boundary layer model [35] allows some surface divergence and predicts the -2/3 power of the Sc for low wind speeds and -1/2 power at higher wind speeds. 

As mentioned previously, even when the flow becomes turbulent in the boundary layer, there exists a thin sub-layer close to the surface in which the flow is laminar. This layer and the fully turbulent regions are separated by a buffer layer, as shown schematically in Figure 7.1. In the simplified treatments of flow within the turbulent boundary layer, however, the existence of the buffer layer is neglected. In the laminar sub-layer, momentum transfer occurs by molecular means, whereas in the turbulent region eddy transport dominates. In principle, the methods of calculating the local values of the boundary layer thickness and shear stress acting on an immersed surface are similar to those used above for laminar flow. However, the main difficulty stems from the fact that the viscosity models, such as equations (7.13) or (7.27),... 

The border diffusion layer model was introduced as an amendment to the film model to present a more realistic description. It accounts for an undefined film thickness, turbulence effects, and the role of molecular diffusion. When the flow is turbulent, the flow around the bubble is split into four sections the main turbulent stream, the turbulent boundary layer, the viscous sublayer, and the diffusion sublayer. Eddy turbulence accounts for mass transfer in the main turbulent stream and the turbulent boundary layer. The viscous sublayer limits eddy turbulence effects so that the flow is laminar and mass transfer is controlled by both molecular diffusion and eddy turbulence. Microscale eddy turbulence is assumed to be dominant in the viscous sublayer. Mass transfer in the diffusion sublayer is controlled almost completely by molecular diffusion (Azbel, 1981). 

The present paper deals with a diffusion model of turbulent boundary layer on a flat plate with small polymer additives. 

Ignatiev V,N. Kuznetsov,B.G. Diffusion model of a turbulent boundary layer with polymer,... 

It is now well established that in a turbulent boundary layer flow the transport of momentum to the wall is intermittent. The events responsible for this locally large instantaneous momentum transport are called sweeps and ejections depending on whether the transport is towards or away from the wall. A variety of models describing the flow field in such events as well as their overall organisation in the main flow have been proposed. Visualisation of ejections show that they are associated with a jet-like and a vortex-like flow field. The relation between the strengthes of these fields are not well known since in order to measure the vorticity in such events,one would need probes which are not available to-day. 

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