Numerical study of unsteady cavitating flow in a three-dimensional axial inducer

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1 See discussions, stats, and author profiles for this publication at: Numerical study of unsteady cavitating flow in a three-dimensional axial inducer Conference Paper March 2011 CITATIONS 0 READS 39 6 authors, including: Rafael Campos-Amezcua Universidad Nacional Autónoma de México 17 PUBLICATIONS 139 CITATIONS SEE PROFILE Farid Bakir Ecole Nationale Supérieure d'arts et Métiers 208 PUBLICATIONS 1,097 CITATIONS SEE PROFILE Sofiane KHELLADI Ecole Nationale Supérieure d'arts et Métiers 142 PUBLICATIONS 424 CITATIONS SEE PROFILE Zdzislaw Mazur-Czerwiec Instituto de Investigaciones Electricas 54 PUBLICATIONS 411 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Prediction of trailing edge noise for low speed axial flow fans View project Counter-rotating turbomachines View project All content following this page was uploaded by Rafael Campos-Amezcua on 02 February The user has requested enhancement of the downloaded file.

2 NUMERICAL STUDY OF UNSTEADY CAVITATING FLOW IN A THREE-DIMENSIONAL AXIAL INDUCER R. Campos-Amezcua 1,*, F. Bakir 2, Z. Mazur-Czerwiec 3, R. Rey 2 1 Department of Non-conventional Energy, Electric Research Institute, Cuernavaca, Mexico. 2 DynFluid Laboratory, Arts et Métiers ParisTech, Paris, France. 3 Department of Turbomachinery, Electric Research Institute, Cuernavaca, Mexico. ABSTRACT This work presents the results of numerical simulation of unsteady cavitating flow in a twoblade axial inducer. First, the analysis was carried out for a two-dimensional blade cascade simulating the axial inducer. Later, the numerical simulation was extended to threedimensional inducer. All calculations were realized in steady state and unsteady state. The main purpose of this study is to explore the local cavitation instabilities, such as alternate blade cavitation and rotating blade cavitation, which can appear in this type of devices. The numerical results show that the flow behaviour in the axial inducer is altered by the appearance of the cavitation near to blade leading edge. These cavitating behaviours change with respect to operation conditions of inducer: flow rate and cavitation levels. The numerical simulation was performed using a commercial code based on a cell-centred finite-volume method. The cavitation model used for calculations assumes a thermal equilibrium between phases. It is based on the classical conservation equations of the vapour phase and a mixture phase, with mass transfer due to the cavitation appearing as a source and a sink term in the vapour mass fraction equation. The mass transfer rate is derived from a simplified Rayleigh Plesset model for bubble dynamics. Nomenclature D diameter Greek Subscript f cav detachment frequency α vapour volume fraction 1, 2 inlet, outlet f rotational frequency β blade angle a axial direction l blade chord length η efficiency B bubble l cav cavitation length γ vapour mass fraction c condensation P pressure flow coefficient cav cavitation P v vapour pressure head coefficient e vaporization Q flow rate ω rotational speed g gas R radius ρ density l liquid St Strouhal number σ cavitation number nom nominal T cycle period σ s surface tension t blade tip t time tot total U tangential velocity v vapour v velocity magnitude Introduction An inducer is a key component of rocket engine turbo pumps, which improves suction performance. Generally, the inducers have a small blade number, and these blade lengths are longer than those of standard axial pump impellers. These wide and long blade passages are hardly blocked by cavitation, so inducers can be operated under very low suction pressure conditions without any deterioration of pumping performance. * rafael.campos@iie.org.mx 1

3 The cavitation inception and development on the inducers depend on many parameters: the blade profile, camber, thickness (Horiguchi et al., 2003), incidence angle and leading edge shape (Bakir et al., 2003), as well as the walls roughness, the upstream turbulence, vapour pressure, air content, etc. While the surface cavitation of vapour bubbles occurs near the blade surface, the vortex cavitation is found in the forms of tip vortex, trailing vortex and secondary vortex. Such vortex cavitation generates a large cavitation cloud with many small cavitation bubbles (Kubota et al., 1989). Computational Fluid Dynamics (CFD) has been extensively used to predict the flow through the inducers under non cavitating conditions. However, because of the physical and numerical challenges associated with cavitation, CFD has only recently started to be used to predict cavitating flows. In the last few years, remarkable progresses have been made in the physical cavitation models. Noguera et al. (1993) and Bakir et al. (1998) have already studied experimentally in cavitating and non cavitating regimes the influence of geometrical parameters such as the shape of the blade leading edge and its sharpening. The combination of numerical and experimental approaches in order to achieve this goal has turned out to be a powerful tool (Bakir et al., 2004, Mejri et al., 2005, 2006 and Campos Amezcua et al., 2008, 2009). This paper presents numerical results of unsteady cavitating flows in a two-blade axial inducer. The calculations were performed using the commercial code Fluent with an extension including a modified turbulent viscosity. The paper is organized as follow: first, physical models and numerical aspects are described briefly. After that, numerical results are presented for two different configurations of the inducer: first, an analysis of a simplified 2D blades cascade where the results show the appearance of alternate blade cavitation and rotating blade cavitation at partial loads and low cavitation numbers. Then, the numerical analysis was extended to 3D inducer. NUMERICAL SIMULATION The non cavitating and cavitating flow through an inducer was modelled for two flow rates at different cavitation conditions. The inducer studied in this paper is showed in figure 1. Its main characteristics are listed in table 1. The working fluid is water at. The liquid and the vapour densities are and, respectively; saturation pressure is and surface tension is. The non condensable gas mass fraction is. The first results were obtained using the RNG turbulence model. Later, calculations were carried out using modified RNG model to take into account both intrinsic and system instabilities. In both cases, wall function was used as near wall treatment. Numerical method The commercial code used for all simulations was Fluent. This code employs a cell centred finite volume method that allows the use of computational elements with arbitrary polyhedral shape. Figure 1. Axial inducer with two blades. 2

4 Convective terms are discretized using the second order upwind scheme. The velocity pressure coupling and overall solution procedure are based on a SIMPLE type segregated algorithm adapted to unstructured grids. The discretized equations are solved using point wise Gauss Seidel iterations, and an algebraic multi grid method accelerates the solution convergence. The convergence criteria in the present numerical analysis were at least of three orders of magnitude drop in the mass conservation imbalance and momentum equation residuals, which are deemed sufficient for most steady flow solutions. A more detailed description of the numerical method is available in Kim et al. (1998). Parameter value Parameter value Rotational speed, N rpm Tip chord length, l t 198 mm Maximal efficiency, max 15.5% Solidity, S t =l t /h 2.52 Nominal flow coefficient, nom tip clearance, t 0.65 Nominal head coefficient, nom Blades number, Z 2 Tip diameter, D t 50 mm Tip blade angle, t1 6 Table 1: Characteristics of inducer blade. Cavitation model The cavitation model used for this study was developed by Singhal et al. (2002). It takes into account all first order effects. The influence of slip velocity between the liquid and the vapour phases was not considering. For the multi phase flow solution, the single fluid mixture model was employed. The mixture model solves the continuity and momentum equation for the mixture, and the volume fraction equation for the secondary phases. The cavitation model consists in solving the standard incompressible Reynolds Average Navier Stokes equations with the use of a conventional turbulence model. The working fluid is assumed to be a mixture of liquid, vapour and non condensable gases. The mixture density ρ is defined by: with, and (2) where α g, α l and α v are the non condensable gases, liquid and vapour volume fraction, respectively; and α = α g + α v is the total vapour volume fraction. The vapour mass fraction is governed by the transport equation given by: where is the velocity vector of the vapour phase, is the effective exchange coefficient, and R e and R c are the vapour generation and condensation rate terms (or phase change rates). The above formulation employs a homogenous flow approach. Using the Rayleigh Plesset equation (4) without the viscous damping and surface tension terms and combining with the continuity equations, the expression for the phase change rate is obtained as: Employing the above equation and ignoring the second order derivative of R B, the simplified equation for vapour transport is obtained as: (1) (3) (4) (5) 3

5 The vapour volume fraction can be related to the bubble number and radius of bubble as: (7) Turbulent effects are taken into account by: (8) The phase change rate expressions are derived from (6) as: where: and are empirical coefficients and is the local turbulent kinetic energy. A more detailed description of the cavitation model is available in Singhal et al. (2002). RNG κ ε model. This model was developed by Yakhot et al. (1986). It uses the Re Normalisation Group (RNG) methods to renormalize the Navier Stokes equations, and take into account the effects of smaller scales of motion. The RNG κ ε model is derived from standard κ ε model. The main difference is the form of the dissipation of the kinetic turbulent energy equation. The turbulence kinetic energy, κ, and its rate of dissipation, ε, are obtained from the following transport equations: and Where η, and P κ represents the production of turbulence kinetic energy. (6) (9) (10) (11) (12) (13) The turbulent viscosity,, is given by: The constants of the model are:,,,. RNG κ ε modified model. This turbulent model is an adaptation from the RNG κ ε model. The modification concerns the reduction, in the low vapour ratio regions, of the effective viscosity,. For this, the mixture turbulent viscosity is given by: where mixture density function,, is given by: This model, proposed by Reboud et al. (1998), allows, in the cases of re entrant jet, the convection of the vapour cloud shedding. The RNG κ ε modified model has been implemented in initial code as a User s Defined Function supplied by Fluent. 4 (14) (15) where (16)

6 Geometrical model and grid generation 2D Inducer blade cascade First, the numerical study was carried out on a blade cascade; the numerical domain has been divided into three sub-domains in order to impose moving mesh conditions. Figure 2 (a) and (b) show the three computational sub-domains of the whole numerical model, defined as: upstream region (A), blades region (B), and downstream region (C). Tangential velocity was imposed in the moving region (B) using a sliding mesh technique, whereas regions (A) and (C) were defined as static regions. Boundary conditions at the domain inlet and outlet were imposed far enough ( ) from the leading and trailing edges, in order to avoid influencing the final results. The used boundary conditions are: 1. Constant velocity at the inlet,. The nominal flow ( ) was defined to correspond to an incidence angle of zero. 2. Constant static pressure at the outlet. Its value decreased to get the desired cavitation conditions. 3. No slip condition at the blades boundaries. 4. Sliding interfaces at the limits between (A) (B) sub-domains and (B) (C) sub-domains. 5. Translational periodic condition was applied for two successive blades. The discretization of the calculation domain was done with a rectangle like structured grid. A grid study was carried out on non cavitating flow. Three different meshes were tested: a coarse mesh (300X50), a fine mesh (500X50), and a refined mesh (650X50). The first coarse mesh presented the backflow at outlet domain because of a very important aspect ratio in this region. The fine and refine meshes presented about the same results. Furthermore, three different lengths of inlet and outlet sub-regions were tested: l 1,2 =5 l, l 1,2 =10 l, and l 1,2 =15 l. In all cases, satisfactory results were obtained in non cavitating flow. But, when the cavitating flow was simulated, both, first and second cases, have presented backflow problems in the outlet of domain. Problems of divergence have been also observed when the outlet boundary was placed close to the blade cascade, mainly for too small sigma values. Therefore, all simulations were carried out on fine mesh (500X50) with inlet/outlet sub-domain length of l 1,2 =15 l. Boundary layer meshing was used to ensure adequate mesh refinement near the walls and thus a small dimensionless factor y +, see figure 2. A 1 mm first cell distance was imposed with a growth rate of 1.2 which allowed values of y + between 6 and 51. a) Blade cascade b) Boundary condition and near-wall mesh resolution Figure 2. Blade cascade corresponding to two-blade inducer. 5

7 3D Inducer A hybrid grid was generated using the pre-processor Gambit, for modelled the inducer en 3D. The computational domain was divided into four sub blocks: a grid block lengthened upstream of the leading edge, a blade to blade region, a grid block extended downstream of trailing edge, and the radial tip clearance modelled by a fourth grid block formed by a ring of thickness and an axial length l rotor, see figure 3. Figure 3. Sub blocks of whole computational domain. First, the blade surface was meshed with triangular cells. The tip region and blade edges were meshed with smaller triangles. The grid block of blade to blade region was filled with tetrahedral cells and the rest of blocks were meshed with prism cells. Conformal grid interfaces were used at the boundary of the regions rotor tip clearance and non conformal grid interfaces were used at upstream rotor regions and rotor downstream regions. For the grid independence study, four computational grids were generated with the same meshing strategy. The numbers of cells in these grids were: (a) , (b) , (c) and (d) Figure 4 shows the surface grid of the rotor and a detail at tip clearance for the grid (c) retained for the calculations. Finally, a grid dependence study at tip clearance was made for: 3, 12 and 25 equidistant cells in radial direction. A constant flow rate was given at inlet boundary. The flow rate was varied from = to = A constant static pressure was given at the outlet boundary. RESULTS Figure 4. Computational grid. 2D Inducer blade cascade Calculations of unsteady cavitating flow were carried out for four partial flow rates over the blade cascade of a two blade inducer. The unsteady cavitating flow was characterized by the head coefficient,, the cavitation number,, and the Strouhal number,, given by: where is the time average of. Various forms and behaviours of cavitation have been observed in the blade cascade. They are influenced by the flow rate and cavitation number. 6 (17)

8 Stable cavitation sheet analysis The first results obtained at a flow rate near nominal conditions, Q=0.97 Q nom, present steady cavitating behaviour. The cavitation region starts to form at the leading edge and the suction side for high values. The vapour sheet is symmetrical on two blades for all times and for all values. This region increases as decreases until the vapour region become large enough to block the flow channel resulting in head break down. When the flow rate decreases to Q=0.81 Q nom, the form of cavitation is different to precedent flow rate. Figure 5 (a) shows the contours of vapour fraction for Q=0.81 Q nom and different values, where it is noticed that the cavitation begins with a very small vapour region at the leading edge and suction side for high values. It is observed that a steady behaviour of the cavitation sheet for values between =0.723 and = This stable behaviour is characterized by a symmetrical cavitation sheets attached to each blade. The cavitation length grows gradually as decreases. When decreases even more, the cavitation sheet area increases and it obstructs the flow channel. At =0.219, the length of cavitation sheet containing 10% of vapour in volume, is approximately 60% of the blade spacing, h, thus, the alternate blade cavitation appears when decreases to =0.174 and this asymmetrical cavitation length continued for = Afterwards, for =0.114, the cavitation sheet becomes steady and the cavitation length are the same for the two blades. Alternate blade cavitation is a phenomenon in which the cavitation length on the blades changes alternately from blade to blade. According to Tsujimoto (2005), the alternate blade cavitation starts to develop when the cavitation length, l cav, exceeds about of the blade spacing. The incidence angle to the neighbouring blade decreases and hence the cavitation length on the neighbouring blade decreases also. Then the incidence angle of the original blade increases and the cavitation length on it also increases. Figure 5 (b) presents the cavitation sheet behaviour for Q=0.60 Q nom as decreases. Analogous to Q=0.81 Q nom, the alternate blade cavitation starts as soon as the l cav /h ratio is higher than 65%, i.e. for = Figure 5. Contours of vapour fraction (α=10%). Unstable cavitation sheet analysis Numerical simulations for low flow rates (below Q=0.55 Q nom ) present many numerical divergence problems. Therefore, the numerical calculations were performed to a lower flow rate of Q=0.56 Q nom where rotating cavitation was observed. For this flow rate, the cavitation sheet has 7

9 different forms as is decreased. Symmetrical cavitation lengths were observed for high values,, similar to higher flow rates cases. Rotating cavitation appears as soon as is decreased to. After that, cavitation length become symmetrical on both blades for all values lower than. Figure 6 (a) shows the contours of vapour fraction ( ) at different times of the rotating cavitation cycle,, for. Monitoring the cavitation sheet evolution on blade 1, it is observed that the cavitation length is the same on both blades at and. At the beginning of the cycle, the cavitation length on blade 1,, decreases with time. So, at, the size of is the smallest on blade 1, while it becomes the largest on blade 2, the cavitation length on blade 2 is inversed to the one on blade 1. So, decreases from to where is the smallest on blade 2 and is the largest on blade 1. The reference time,, has been defined as the time for one impeller revolution, i.e.. The sheet cavitation has a cyclic unsteady behaviour with low frequency equal to on one blade. The variations of cavitation forms in time change the flow dynamics which cause pressure fluctuation upstream. The frequency analysis on the absolute frame gives because of cavitation detachment on two blades. (a) RNG κ-ε model (b) RNG κ-ε modified model Figure 6. Rotating cavitation behaviour on two-blade inducer at =0.258 and Q=0.56 Q nom. Coupling of the instabilities and the self oscillation of a cavitating sheet Numerical simulations were completed using the RNG modified model for and. This modification allows the interaction between the unsteadiness of the two blades and the self oscillation of cavitating sheet. Figure 6 (b) presents the contours of vapour fraction at different times on a blades cascade. The cavitating sheet shows a quasi cyclical unsteady behaviour with a detachment frequency equal to on relative frame and on absolute frame. 8

10 The cavity has a similar cyclical unsteady behaviour as in analysis using RNG model, but now, the results show the vapour sheet detachment ( to, blade 1 in Figure 6 (b)), followed by its convection downstream ( to, blade 1) and then, the cavitation passing from blade 1 to blade 2 at blades cascade throat ( to ). The curves of figure 7 show the cavitation length in time,, calculated on each blade based on a vapour fraction of (see Figure 6 (a) ). A negative sheet length means that a cavitation sheet was attached to the pressure side of the blade. It appears when the cavitation length of neighbouring blade is large enough which produces the blockage of the channel flow. Thus, the flow passes only through the other channel. This figure also shows cavitation length evolution for previous results (RNG model). The four curves have a similar behaviour but the cavitation length is larger with the modified RNG compared to the standard model. The local length fluctuations observed on modified turbulence model are caused by the self oscillation of cavitation region. Figure 7. Comparison of temporal evolution of cavitation length on two blade inducer. Calculation using RNG model and RNG modified. 3D Inducer Figure 8 shows, in front view, side view and isometric view, the temporal evolution of the isosurface of vapour ( ) in the two-blade inducer. Twenty-four instants can be observed between and. The pictures were obtained when all flow parameters were stabilized, the time step used for all numerical simulations was. This figure shows that at, the cloud cavitation has the shape of a crown at the periphery of the inducer. This cavitation has also the shape of the torch to upstream of the inducer. For this moment, the cavitation length is symmetrical on both blades. The cavitation cloud located on the periphery and the cavitation torch are connected by a narrow region of vapour formed along the leading edge, from the tip blade to the inducer hub. Later, at, cavitation develops gradually on the leading edge of blade 2 (green blade) that is bigger than that on the blade 1 (blue blade). Then the cavitation cloud decreases gradually on the leading edge of blade 2 until. In contrast, the cavitation cloud on blade 1 is the biggest. Finally, the cloud cavitation begins to grow on blade 2 and it decrease on the 1 until these become the biggest and the smallest respectively at. 9

11 a) front view b) Side view c) Isometric view Figure 8. Temporal evolution of iso-vapour ( in the two-blade inducer as and. Therefore, this fluctuation of cavitation size is almost cyclic. As can be seen in the figure, the cavitation length is maximal at, and on the blade 2 and,, and on the blade 1. Thus, the cavitation fluctuation period is and its frequency is. The fluctuations can be driven by the cavitation torch formed upstream of the inducer. T torch run in the direction of the inducer rotation. So the torch turns 1 time while the inducer turns 9 times. First, the unsteady cavitating calculations were performed for low cavitation numbers. Calculations were realized, for, from the results obtained at, and then, for. This last calculation corresponds to over 10% of the inducer pressure drop. As can be seen in figure 9, the cavitation cloud is bigger on the blade 1 (blue blade) than on the blade 2 (green blade), at. Then this vapour region decreases gradually and it becomes smaller on the blade 1 while it is bigger on the blade 2. The figure shows three successive cycles of this cavitation fluctuation which occur for 22 inducer cycles. Thus, the cavitation fluctuation period is and its frequency is. a) front view b) Side view c) Isometric view Figure 9. Temporal evolution of iso-vapour ( in the two-blade inducer as and. CONCLUSIONS Unsteady numerical simulations were carried out over two different configurations: first, in a blade cascade of a two-blade inducer, then the calculations were realized for a 3D geometry of same inducer. 10

12 Cavitating flow in two blade inducer, for various values and flow rates, predicted three types of cavitation behaviour on the blade cascade: 1. stable behaviour with symmetrical cavitation length, 2. stable behaviour with non symmetrical cavitation length, 3. cyclical unstable behaviour with non symmetrical cavitation length. Cavitation length behaviour was symmetrical and stable for a high flow rate of. Alternate blade cavitation was observed for lower flow rates, when the ratio was higher than about. Finally, the rotating cavitation was observed only for a partial flow rate of, where the calculations were carried out using RNG model and RNG modified model. Numerical results showed three different mechanisms of cavitation instabilities: 1. Self oscillation of the cavitation sheet due to the interaction between the recirculation flow and the cavity surface in the venture geometry. 2. Rotating cavitation due to the interaction of the sheet cavitation in a blade with the leading edge of the neighbour blade in blade cascade. 3. Coupling of the rotating cavitation and the self oscillating of the cavitation sheet in blade cascade. Finally, the unsteady cavitating calculations realized for the inducer 3D have highlighted the difficulty in obtaining numerical results and for compiling and analyzing them. The results show that rotating cavitation appears on 3D geometry but it is less obvious than on the blades cascade. The shape and behaviour of cavitation is greatly disturbed by the radial clearance, which also modifies the torch which is formed upstream of the inducer. References [1] Bakir, F., Kouidri, S., Noguera, R., and Rey, R. (1998) Design and analysis of axial inducers performance, ASME Fluid Machinery Forum, FEDSM , Washington, D.C. [2] Bakir, F., Kouidri, S., Noguera, R., and Rey, R. (2003) Experimental analysis of an axial inducer: influence of the shape of the blade leading edge on the performance in a regime of cavitation, J. Fluids Eng. Vol. 125, pp [3] Bakir, F., Rey, R., Gerber, A. G., Belamri, T., and Hutchinson, B. (2004) Numerical and experimental investigations of the cavitating behaviour of an inducer, Int. J. Rotating Mach., 10(1), pp [4] Campos-Amezcua, R., Bakir, F., Khelladi, S., Rey, R. (2009) Numerical study of the unsteady cavitation flow, 12 th Int. Symp. on Transport Phenomena and Dynamics of Rotating Machinery, February 17 22, Honolulu, Hawaii. [5] Campos-Amezcua, R., Bakir, F., Khelladi, S., Rey, R. (2009) Numerical analysis of unsteady cavitating flow in an axial inducer. J. Power and Energy. Vol. 224, pp [6] Horiguchi, H., Semenov, Y., Nakano, M., and Tsujimoto, Y. (2003) Linear stability analysis of the effects of camber and blade thickness on cavitation in inducers, 5 th Int. Symposium on Cavitation, OS 4 004, November 1 4, Osaka, Japan. [7] Kim, S.E., Mathur, S.R., Murthy, J.Y., and Chouhury, D. (1998), A Reynolds Average Navier Stokes solver using unstructured mesh based finite volume scheme, Proc. 36 th AIAA Aerospace Sciences Meeting and Exhibit, AIAA , Reno, NV. [8] Kubota, A., Kato, H., Yamaguchi, H., and Maeda, M. (1989) Unsteady structure measurement of cloud cavitation on a foil section using conditional sampling technique, J. Fluids Eng. Vol. 111, pp [9] Mejri, I., Bakir, F., Kouidri, S., and Rey, R. (2005) Influence of peripheral blade angle on the performance and the stability of axial inducers, J. Power and Energy. Vol. 219, pp

13 [10] Mejri, I., Bakir, F., Rey, R., and Belamri, T. (2006) Comparison of computational results obtained from a homogeneous cavitation model with experimental investigations of three inducers, J. fluids Eng. Vol. 128 (6), pp [11] Noguera, R., Rey, R., Massouh, F., Bakir, F., and Kouidri, S. (1993) Design and analysis of axial pumps, ASME Fluids Engineering, Second Pumping Machinery Symposium, pp , Washington, USA. [12] Reboud, J., Stutz, B. and Coutier-Delgosha, O. (1998) Two-phase flow structure of cavitation: Experiment and modelling of unsteady effects. 3 rd Int. Symposium on Cavitation. April 7-10, Grenoble, France. [13] Singhal, A. K., Athavale, M. M., Li, H. Y., and Jiang, Y. (2002) Mathematical basis and validation of the full cavitation model, J. Fluids Eng. Vol. 124 (3), pp [14] Tsujimoto, Y., Horiguchi, H., and Qiao, X. (2005) Backflow from inducer and its dynamics, ASME Fluids Eng. Conf. 5 th Pumping Machinery Symp., June 19 23, Houston, Texas. [15] Yakhot, V. and Orszag, A.S. (1986) Renormalization group analysis of turbulence Basic theory. J. Scientific Computing. Vol. 1 (1), pp View publication stats