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Print version ISSN 0100-7386
J. Braz. Soc. Mech. Sci. vol.22 n.1 Rio de Janeiro 2000
C. J. Deschamps
A. T. Prata
R. T. S. Ferreira
Federal University of Santa Catarina, Department of Mechanical Engineering, 88040-900, Florianopolis, SC, Brazil
The work considers the modeling of turbulent flow in radial diffuser with axial feeding. Due to its claimed capability to predict flow including features such as separation, curvature and adverse pressure gradient, the RNG k-e model of Orzag et al. (1993) is applied in the present analysis. The governing equations are numerically solved using the finite volume methodology. Experiments were conducted to assess the turbulence model. Numerical results of pressure distribution on the front disk surface for different flow conditions when compared to the experimental data indicated that the RNG k-e model is adequate to predict this class of flow.
Keywords: Turbulence, RNG k-e model, Radial Diffuser.
A three dimensional schematic view of a radial diffuser and the geometric parameters that govern the flow are shown in Figs. 1a and 1b, respectively. The fluid enters the diffuser flowing axially through the feeding orifice, hits the front disk, and after being deflected by it a radial flow is established. The impact of the flow on the front disk produces a bell-shape pressure distribution and, depending on the gap between the disks and on the flow Reynolds number, negative pressure regions may occur.
The radial diffuser geometry of Fig. 1 represents a basic configuration for many engineering flows. Examples include reed type valves of reciprocating compressors (Prata and Ferreira, 1990), aerostatic bearings (Hamrock, 1994), electro-discharge machining (Osenbruggen, 1969), aerosol impactors (Marple et al., 1974), injection molds used for polymer processing (Pearson, 1966), pilot valves employed in hydraulic and pneumatic components (Hayashi et al., 1975), and prosthetic valves used to replace diseased natural heart valves in humans (Mazumdar and Thalassoudis, 1983).
Despite the numerous works related to laminar radial flows, very little attention has been given to turbulent flows. For references on laminar radial flow the interested reader is referred to Hayashi et al. (1975), Wark and Foss (1984), Ferreira et al. (1989), Prata et al. (1995), Possamai et al. (1996), and the literature cited therein.
The few works dealing with radial turbulent flow focused on pure radial flow between parallel disks without considering the inlet (Ervin et al., 1989; Tabatabai and Pollard, 1987). Apparently, the first attempt to solve the turbulent flow in axially feeding radial diffusers was made by Deschamps et al. (1988). There it was found that the high Reynolds number k-e model (Launder and Spalding, 1974) used to close the averaged Navier-Stokes equations is unable to predict the flow, even with the inclusion of correction terms to take into account effects such as flow curvature. The poor quality of the numerical solution was linked to the wall-functions needed in the model. This was confirmed later when a low Reynolds number k-e model (Jones and Launder, 1972), which does not use wall-functions, produced better flow predictions (Deschamps et al., 1989). Nevertheless, even for this model there were significant differences between experiments and computations, which were attributed to the prediction of excessive turbulence levels at the entrance of the radial diffuser.
The main goal of the present paper is the numerical solution with experimental validation of the turbulent flow through the geometry depicted in Fig. 1. Due to the claimed capability to predict flows that include features such as stagnation and separation regions, curvature and adverse pressure gradients (all of them present in the flow considered here) the RNG k-e model of Orzag et al. (1993) was adopted in this work.
Experimental Setup and Procedure
A careful experimental setup was built to measure the pressure distribution on the front disk as a function of the Reynolds number, Re (= rUind/m ), and the displacement between the disks, s.
The airflow in the test section was supplied by a compressed air reservoir through a long horizontal aluminum tube. A control valve placed at the entrance of the tube allowed adjustments of the flow rate, which was measured by an orifice flow meter. Prior to reach the diffuser the flow entered a feeding orifice whose diameter d was equal to the diameter of the aluminum tube. Two diameter ratios between the front disk and the feeding orifice were tested (D/d=1.45 and 3).
The test section consisted of a radial diffuser with a high-precision positioning system for the front disk to allow the measurement of the gap between the disks. Along the front disk diameter it was inserted a sliding bar provided with a small tap hole (0.7mm diameter) and an internal connecting perforation (2 mm diameter) to one of its extremities, where an inductive pressure transducer collected the pressure signal. At the other extremity of the bar an inductive displacement transducer was placed to allow the accurate location of the pressure tap. Both, pressure and location data were then fed into a data acquisition system.
Of paramount importance in the experiments was the correct adjustment of the displacement to the desired value due to the strong influence of this parameter on the flow field. At the beginning of each experimental run the first step was to position the front disk at the desired location, by making adjustments concerned with parallelism and concentricity of the disks. Next, a sample data acquisition was processed and through a visual observation of the symmetry of the pressure profile along the front disk, the disks parallelism and concentricity was checked. After the correct positioning of the system was met the front disk was set to the desired displacement s/d. To initiate the experimental run the control valve was adjusted to the required Reynolds number with the sliding bar positioned so that pressure tap was at the front disk rim. At this point everything was ready for the data run. The sliding bar was carefully moved from one side to the other, with the acquisition data system registering the pressure distribution simultaneously.
The uncertainty associated to the experimental data is less than 5%. A more detailed account of the experimental setup and procedure can be found in Possamai et al. (1995).
The most frequent approach in turbulence modeling is the time/ensemble averaging of the flow transport equations, where the turbulent component of any property is defined as the departure from its time/ensemble averaged value. For an isothermal and incompressible flow under the effect of no body force the equations of motion (here written in Cartesian tensor notation) are:
Momentum conservation for the Ui component of velocity,
The term appearing in Eq. (2) is the Reynolds stress tensor and is never negligible in any turbulent flow. Equations (1) and (2) can only be solved if the Reynolds stresses are known, a problem referred to as the 'closure problem' since the number of unknowns is greater than the number of equations.
By manipulating the Navier-Stokes equation for instantaneous velocity it is possible to obtain a transport equation for the Reynolds stress . However, the resulting set of equations is not closed since they include higher order correlations, such as the third order moments , as well as correlations between fluctuating velocities and pressure. Any attempt to provide transport equations for these higher order correlations leads to the appearance of even higher correlations and, consequently, to a dramatic increase in the computing time required to calculate flows. For this reason, most works using this level of modeling have used transport equations for second order correlations and simple algebraic expressions to approximate the triple-moments and the correlations between fluctuating velocities and pressure. Models for closing the Reynolds stress transport equations, following the aforementioned procedure, were proposed quite early (see for instance Rotta, 1951).
A considerable obstacle for the use of Second Moment closures is the modeling of the near-wall region and, despite much effort have been directed to the solution of the problem, progress has not reached the point for the full benefit of flow in complex geometries as the one considered here. On the other hand, the employment of wall-functions, to bridge the whole of the near-wall sublayer where viscous effects are significant, is unsuitable for the present flow situation, as hinted at by previous works (Deschamps et al., 1989). This is mainly related to the numerical aspect and caused by the condition of minimum turbulence level that must be observed in the numerical solution next to the wall; for practical reasons usually fixed as y+ » 11.6, where y+ = r u* y/m and u* = (tw./r )1/2. Naturally, in flows where important features occur very close to the wall it is quite difficult if not impossible to balance the needed grid resolution against the minimum value for y+. Given the foregoing reasons and in order to predict the flow in radial diffusers it seems to be essential to avoid the use of wall-functions and to include the near-wall region in the calculations.
An alternative for the turbulence modeling follows the Boussinesqs hypothesis of a 'turbulent' or 'eddy' viscosity nt and which considers that turbulence is proportional to the velocity gradient acting like the viscous stresses. Kolmogorov proposed a generalized form of this hypothesis as follows:
where dij is the Kronecker delta and the kinetic energy of the turbulent motion, k, is defined as k=()/2. Substitution of Eq. (3) into Eq. (2) results in the averaged Navier-Stokes equations with the Reynolds stresses modeled via the viscosity concept:
where neff = n + nt. By far the most common choice for calculating of nt has been that in terms of the turbulence kinetic energy k and its rate of dissipation, e, i.e.
Models of this kind were originally proposed by Harlow and Nakayama (1968) and subsequently refined by Launder and Spalding (1972). Later, Jones and Launder (1972) included low-Reynolds-number effects into the k-e model (by making certain coefficients dependent on the turbulent Reynolds number) so that it can be used to compute near-wall flows as well as those where wall effects are not present.
Due to its robustness, economy and acceptable results for a considerable amount of flows the k-e model has been the most used model for numerical predictions of industrial flows. However, it is known to have deficiencies in some situations involving streamline curvature, acceleration and separation; all of them are present in the case of flow through radial diffusers. For instance, turbulence is very sensitive to small amounts of curvature of the streamlines; see Bradshaw (1973). The effects of curvature tend to increase the magnitude of the turbulence shear stress where the angular momentum of the flow decreases in the direction of the radius of curvature, and to decrease when the angular momentum increases with the radius. Hence, for example, on a typical turbomachine blade the skin friction may be reduced by the curvature on the convex surface by as much as 20% and increased on the concave surface by a comparable amount. Such effects cannot be accounted for in turbulence models based on the simple eddy-viscosity hypothesis unless some ad hoc extra terms be introduced into the equations (it is opportune to point out that the Reynolds stress equations, in the case of curved streamline flows, contain in exact form extra-strain production terms that account for the curvature). Moreover, in the presence of adverse pressure gradient regions, the equation for e is known not to be capable of responding to the surge of kinetic energy, returning an excessive level of turbulence that can lead to a delay of an eventual flow separation or even to a total suppression of it, as pointed out by Rodi and Scheuerer (1986).
Orzag et al. (1993) have derived a new form of k-e model from the governing equations for the fluid motion using Renormalization Group (RNG) methods. The novelty of the so called RNG k-e model, compared to the standard k-e model, is that constants and functions are evaluated by the theory and not by empiricism and that the model can be applied to the near-wall region without recourse to wall-functions or ad-hoc function in the transport equations of the turbulence quantities. Due to this mathematical foundation, compared to the semi-empirical approaches adopted in the standard k-e, Orzag and his colleagues argue that the RNG k-e model offers a wider range of applicability. Some examples of flows where the RNG k-e model has been seen to return better predictions than the standard k-e are those including flow separation, streamline curvature and flow stagnation. As pointed out before, all these flow features are present in the case of radial diffusers and, therefore, it seemed natural to adopt the RNG k-e model in the present work.
The effective viscosity in the RNG k-e model is given by
where Cm = 0.0845. Equation (6) is valid across the full range of flow conditions from low to high Reynolds numbers. The turbulence kinetic energy k and its dissipation e appearing in Eq. (6) are obtained from the following transport equations:
where the values of Ce1 and Ce2 are equal to 1.42 and 1.68; respectively. The inverse Prandtl number a for turbulent transport is given by the following relationship:
with ao = 1.0. The rate of strain term, R, is given by
where h = Sk / e, ho » 4.38, b = 0.012 and S2 = 2Sij Sij in which Sij is the rate of strain tensor. In regions of small strain rate, the term R has a trend to increase neff somewhat, but even in this case neff still is typically smaller than its value returned by the standard k-e model. In regions of elevated strain rate the sign of R becomes negative and neff is considerably reduced. This feature of the RNG k-e is responsible for substantial improvements verified in the prediction of large separation flow regions. Finally, the reduced value of Ce2 in the RNG theory, compared to the value of 1.9 used in the standard k-e turbulence model, acts to decrease the rate of dissipation of e , leading to smaller values of neff.
In order to solve Eqs. (4), (7) and (8) boundary conditions are required at inlet, walls, axis of symmetry and outlet. For small values of s/d Ferreira et al. (1989) recognized that as the flow exits the feeding orifice and enters the diffuser the reduction of the passage area brings about a strong flow acceleration next to the orifice wall. Due to this phenomenon the inflow velocity profile at the feeding orifice plays no role in the solution of the flow field in the diffuser. Therefore, the inlet boundary conditions for the axial and radial velocity components were specified as U=Uin and V=0, respectively, with Uin being the average velocity in the orifice. Although no information is available for the turbulence kinetic energy, numerical tests indicated that when the level of the turbulence intensity I, based on Uin, is increased from 3% to 6% no significant change is observed in the predicted flow. Therefore, an intensity of 3% was used in the calculation of all results shown in this work. Finally, the distribution of the dissipation rate was estimated based on the assumption of equilibrium boundary layer, that is
with the mixing length, , being calculated using an empirical coefficient for turbulent pipe flow, that is, =0.07d/2.
At the solid boundaries the condition of no-slip and impermeable wall boundary condition were imposed for the velocity components, that is, U=V=0, with calculations being extended up to the walls across the viscous sublayer. For the turbulence quantities k and e rather than prescribing a condition at the walls, they were calculated in the control volume adjacent to the wall following a two-layer based non-equilibrium wall-function. In the plane of symmetry, the normal velocity and the normal gradients of all other quantities were set to zero.
At the outlet boundary two different procedures had to be adopted. For D/d = 3 the diffuser exit is far enough downstream and a condition of parabolic flow can be assumed. Yet, for the much smaller ratio, D/d = 1.45, this is not possible and, therefore, the solution domain had to be extended well beyond the diffuser exit and the atmospheric pressure verified in the experiment was set to the outlet. The boundary condition for k in this case was fixed according to a turbulence intensity of 3% whereas the dissipation rate was estimated based on the same assumption of equilibrium boundary layer used at the orifice inlet, Eq. (11). Given the wall jet characteristic of the flow exiting the diffuser it is expected that any eventual inaccuracy of the above outlet conditions will not have a significant impact on the numerical solution.
The numerical solution of the governing equations was performed using the commercial computational fluid dynamics code FLUENT, version 4.2 (1993). In this code the conservation equations for mass, momentum and turbulence quantities are solved using the finite volume discretization method (Patankar, 1981). For this practice the solution domain is divided in small control volumes, using a non-staggered grid scheme, and the governing differential equations are integrated over each control volume with use of Gauss' theorem. The resulting system of algebraic equations is solved using the Gauss-Seidel method and the SIMPLE algorithm.
In the finite volume method, interpolation of properties at the control volume faces can be of primary importance on the accuracy of the numerical results. The classical approach of first order accurate upwind differencing usually suffers from severe inaccuracies in complex flow situations originated by truncation errors and streamline-to-grid skewness. A consequence of the first is that the only truncation-error-free problems are those whose solutions vary almost linearly with the grid index in the streamwise direction. The second source of error occurs where the vector velocity is not aligned with the grid lines (as in recirculating flow regions), and usually is referred to as false diffusion. Recirculating regions of course are a common feature in radial diffusers and, therefore, such flow situations are susceptible to this sort of error. An effective approach to reduce truncation error, while still maintaining the grid size within computational resource limits, is the introduction of a more accurate differencing scheme into the numerical analysis. In the present work, the QUICK scheme was adopted in the solution of the momentum equations, yielding a second order accuracy for the interpolated values. Yet, for the transport equations of turbulence quantities the Power Law Differencing Scheme (PLDS) of Patankar (1980) was adopted since the unboundedness of the QUICK scheme usually introduces serious numerical instabilities, causing calculations to diverge. Nevertheless, there is some evidence (Craft, 1991) that in the case of these equations the source terms are dominant, with the convective terms playing secondary role.
Two grid levels (70x80 and 100x140, axial x radial) were used to assess the numerical truncation error. The refinement was mainly promoted in the entrance of the diffuser, where flow property gradients are steeper. Of great help to this test was some evidence of the discretization needed for the analysis and made available by Deschamps et al. (1989) and Possamai et al. (1995).
Because of the strong non-linearity of the equations, under relaxation coefficients were required. For the velocity components these coefficients were 0.15, for pressure 0.25 and for the turbulence quantities 0.2. At the very first interaction of the numerical procedure the sum of the residual of all algebraic equations (all five variables included) was on the order of one; convergence was stopped when this sum was less than 5x10-4.
Results and Discussions
The flow through the radial diffuser in Fig. 1 was investigated for three displacements, s/d (=0.05, 0.07 and 0.10), three Reynolds number, Re (=10,000; 20,000 and 40,000), and two diameter ratios, D/d (=1.45 and 3). In all cases, the feeding orifice length was e/d=1.0.
The numerical solution was validated by means of sensitivity tests of the results with respect to grid refinement and boundary conditions. The numerical solution is expected to represent thus a pure prediction of the flow through the turbulence model, for which an assessment was possible by a comparison between numerical results and experimental data. The numerical validation and the turbulence model assessment were conducted with reference to results of dimensionless pressure P*(=2p/r Uin2) on the front disk surface.
Figure 2 shows the result emerging from the sensitivity tests. In Fig. 2a results of radial pressure distribution on the disk surface yielded by two different levels of grid refinement (70x80 and 100x140) are compared for the flow situation D/d=3 and Re=20,100. It is clear that both results are virtually the same and hence, in order to reduce computing times, the less refined grid was used in the remaining calculations. Another source of uncertainty that had to be investigated is that related to the inlet boundary condition for turbulence at the entrance of the feeding orifice since the experimental setup used in this work cannot provide such information. Because the flow upstream of the test section follows a straight smooth pipe, it was assumed that levels of turbulence are moderate and therefore the levels of turbulence intensity I tested were 0.03 and 0.06. The result of the test (Fig. 2b) shows that the pressure distribution on the disk surface was not significantly affected by the variation in the level of turbulence intensity. A value of I = 0.03 will be adopted for the remaining calculations.
Figures 3 and 4 show the radial pressure distribution along the front disk surface obtained from the experiments and computations for a variety of flow geometries. In all situations the pressure profile exhibits a plateau on the central part of the curve (r/d < 0.5), as previously verified for the laminar flow by Ferreira and Driessen (1986) and Ferreira et al. (1989). Also similar to the laminar flow is the sharp pressure drop at the radial position r/d » 0.5, which is due to the change in the flow direction. For the outer part of the curve (r/d > 0.5) the pressure level never recovers a positive value, a situation which is also verified in the laminar flow for combinations of both large displacement and high Reynolds number.
For the diameter ratio D/d=3 (Figs. 3a and 3b) the pressure level goes to almost zero for regions r/d>1.25. This is not the case for the smaller diameter ratio D/d=1.45 where the pressure even at the exit of the diffuser is seen not to reach the atmospheric condition, but still remains negative. The reason for this being that in the latter case the diffuser does not have sufficient length to allow for complete pressure recovery, which does not defy expectation.
The good agreement between experiments and computations seen in Figs. 3 and 4 provided confidence in the turbulence model. Thus, the next step in the analysis was to generate numerical simulations for flow situations not included in the experimental investigation. The computations were then conducted for D/d=1.45, considering three displacements (s/d=0.05, 0.07 and 0.10) and two Reynolds numbers (Re=10,000 and 40,000). The results plotted in Figs. 5a and 5b at first sight show no significant difference between the pressure distributions on the valve surface for the two Reynolds numbers explored. However, a first distinction between the curves is that for increasing Re values, the magnitude of the negative pressure profiles decreases. To support the explanation for this feature Fig. 6 was prepared. In this figure dimensionless stream-function y * (= y /m) contours are plotted at the entrance of the diffuser for s/d=0.05 and 0.10 at two flow rate conditions (Re=10,000 and 40,000). For the smaller value of s/d the flow is seen to separate at r/d » 0.5 and to reattach downwards inside the diffuser. As the gap between the disks is increased to 0.10, and the flow inertia becomes stronger, the separation region is increased and the recirculating zone moves into the diffuser exit. Since the negative pressure values are dictated by the flow passage area in the diffuser, the rise in those values with increasing values of Re is a direct consequence of the growth of the separated flow region in the diffuser.
Another important detail of Fig. 5 is disclosed with the help of Fig. 7. There the pressure distributions for Re = 10,000 and 40,000, normalized by the pressure value at the center of the front disk (r/d=0), are presented for s/d =0.05 and 0.10. The figure shows that the pressure drop at r/d » 0.5 is steeper for smaller displacements. This is an expected result since as the gap between the disks increases, the change in the flow direction at r/d » 0.5 becomes less stiff. Additionally, for s/d=0.05 an increase in the Reynolds number brings about a considerable enhancement of the negative region in the pressure distribution, whereas, for s/d=0.10, the Reynolds number effect in the shape of the pressure distribution is much less prominent. This feature is related to the size of the separated flow region in the diffuser, as can be noticed from Fig. 6.
The present work has presented a numerical and experimental investigation of the incompressible turbulent and isothermal flow in a radial diffuser. This is the basic flow problem associated with several engineering flows, such as automatic valve reeds of reciprocating compressors, aerostatic bearings and aerosol impactors. The flow was analyzed for different parameters such as Reynolds number, diameter ratios and gap between the disks.
The RNG k-e turbulence model used to predict the flow was found to reproduce well the experimental results. It should be mentioned that a complete assessment of the turbulence model would require comparisons between numerical results and experimental data of turbulence quantities, such as Reynolds stresses. This could not be addressed in the present work due to limitations of the experimental setup.
One of the main features observed in all flow situations is the presence of a separated flow region in the diffuser. This contributes greatly to the negative pressure region observed along the entire diffuser on the front disk surface. For the cases investigated here, it seems that as the gap between the disks is increased the shape of the pressure distribution on the disk surface becomes less and less dependent on the Reynolds number and the gap itself.
This work is part of a technical-scientific cooperation program between Federal University of Santa Catarina and the Brazilian Compressor Industry, EMBRACO. Partial support from the Brazilian Research Council, CNPq, is also appreciated.
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Article presented at the 1st Brazilian School on Transition and Turbulence, Rio de Janeiro, September 21-25, 1998.
Technical Editor: Atila P. Silva Freire.