CFD analysis of the transient turbulent flow in sliding gate valves

Emilio E. Paladino

ESSS – Engineering Simulation and Scientific Software, Florianopolis, SC, Brazil Marcus Reis

ESSS – Engineering Simulation and Scientific Software, Florianopolis, SC, Brazil Jorivaldo Medeiros

CENPES – Research and Development Center – PETROBRAS, Rio de Janeiro, RJ, Brazil

Clovis R. Maliska SINMEC – Computational Fluid Dynamics Lab – Federal University of Santa

Catarina, Florianopolis, SC, Brazil Aristeu da Silveira Neto

LTCM – Heat and Mass Transfer Lab – Federal University of Uberlandia, Uberlandia, MG, Brazil

Abstract

This work presents an analysis of the transient turbulent flow within flue-gas valves. These sliding-gate-type valves are used for pressure control of an FCC (Fluidized Catalytic Cracking) unit regenerator. Downstream these valves, refractory layer damage have been taking place during unit operations and it is supposed to be caused by fatigue due to pressure oscillations along the walls. The purpose of the CFD model is to calculate the oscillating mechanical forces due to fluctuating pressure field and use them as input for structural dynamic analysis using ANSYS. Two turbulence modeling approaches are used to study the pressure fluctuations in these valves: the RANS (Reynolds Averaged Navier-Stokes Equations) and the LES (Large Eddy Simulation) approaches. The latter is supposed to be capable of capturing higher frequencies. This is a required feature of the model, as some experimental measurements using accelerometers located externally to the wall, have shown accelerating frequencies as high as 250 Hz. Thus, the model used has to be capable to predict these frequencies. Nevertheless, the RANS k-ε approach is also used in order to evaluate differences between the turbulence approaches and to compare the capabilities of each one in the context of this research, since the k-ε approach demands less computational efforts. The simulations are carried out using CFX 5.6 solver. Some preliminary models for the structural analysis are also presented.

Introduction Two sliding gate valves are used to control the pressure at the exhausting duct of an FCC (Fluidized Catalytic Cracking) regenerator unit. These valves introduce a very large area contraction, generating strong local accelerations. The problem consists of some breakdown/failure points of the refractory layer at these exhausting tubes in some PETROBRAS refineries during normal operating conditions. Preliminary analysis has indicated that these failures are due to fatigue, i.e., oscillatory repetitive loads over the refractory layer, mainly due to the fluctuations of the flow pressure. This work presents the preliminary results of a CFD analysis performed aiming at determining and modeling the origin of such loads. The final objective of the work is to perform a one-way fluid structure interaction analysis, i.e., to evaluate the transient fluid forces over the structure. As a second stage the goal is to perform a structural modal analysis

and compare the response of the wall transient displacements with the measured ones at tube wall in real operating conditions.

Initially, no information about the origin of these loads was available. The only field information was the acceleration measurements at the external tube wall. The data was obtained using an accelerometer on the external wall at the refractory failure points and processed through FFT (Fast Fourier Transform) analysis in order to obtain the peak frequencies at which the maximum displacements are located. Figure 1 (from Medeiros et al.(2000)), shows the acceleration spectrum obtained. It is important to emphasize the presence of some peaks at frequencies between 150 to 250 Hz.

Figure 1 – Spectral acceleration at the tube wall. Field measurements

The exhausting gas (known as flue gas) consists of a mixture of N, CO2, H2O and small fractions of other gases, with a minor fraction of solid catalyst particles (residuals from cyclone separation). The gas speed is high (~350 m/s) and so are the temperatures (~940 K). Because of these extreme operating conditions, measurements of fluid dynamic parameters such as pressure and velocity fluctuations are very difficult to be performed. Furthermore, as it will be seen in the results, local pressure transient signal would not provide any information about the origin of high frequency loads, and integrals of pressure along the walls have to be performed in order to obtain forces acting on the structure.

Modeling approach A tri-dimensional transient CFD model has been implemented to obtain transient pressure loads acting on the structure. As it has already been mentioned, two approaches were considered: the U-RANS – Unsteady Reynolds Averaged Navier-Stokes using the standard k-ε (Launder & Spalding (1974)) model and the Large Eddy Simulation (LES) approaches.

To obtain the required information (high frequency fluctuations) the LES approach has shown to be the most adequate. Nevertheless, a RANS k-ε simulation was performed to compare and evaluate if this lower computationally costly approach could provide some basic results. The next sections will give some details of both approaches.

p p q

0

1

2

3

4

5

6

7

8

9

10

0 100 200 300 400 500

f = 220 Hz

ω [Hz]

Acc

eler

atio

n [m

2 /s]

Averaged and Filtered equations The instantaneous mass, momentum and energy conservation equations are,

( ) ( ) 0=⋅∇+∂∂ Uρρt

(1)

( ) ( ) ( )( ) fUUUUU ρμρρ Tpt

∇+∇⋅∇+−∇=⋅∇+∂∂

(2)

( ) ( ) ( )Thht TT ∇∇=∇+∂∂ λρρ U (3)

In equation (3) hT represents the total enthalpy, given by,

2

21 UhhT += (4)

and h=h(p,T). Thus, the energy equation takes into account the enthalpy variation due to fluid acceleration and it is valid for high speed flows. The compressibility was considered because in some regions Mach number reaches values of about 0.6. The flue-gas was considered as ideal gas and so, h(p,T)=CPT.

The above equations represent the fluid flow at any Reynolds number including all turbulence effects. These equations could be, a priori, numerically solved and they could provide the whole turbulent spectrum, giving pressure and velocity fluctuations for any frequency. This approach is the so called Direct Numerical Simulation (DNS). For high Reynolds number problems, the computational efforts using the DNS approach would be impracticable with today’s available computers.

The most common approach for modeling industrial turbulent problems is to solve the Reynolds Averaged Navier Stokes Equations. After time averaging the above equations, we have,

( ) 0=⋅∇ Uρ (5)

( ) fUUUU ρμρ +⎟⎠⎞

⎜⎝⎛ ⎟⎠

⎞⎜⎝⎛ ∇+∇⋅∇+−∇=⋅∇

Tp (6)

( ) ( )ThT ∇∇=∇ λρU (7)

Note that, as a consequence of the temporal averaging process, the transient terms are all zero. This new equation set could be solved to obtain the averaged variables, but it is observed the appearance of averages of products in the advective terms. To solve this problem, Reynolds (1883) apud Silveira Neto (2003) proposed the decomposition of instantaneous variables in an averaged part and a fluctuating part as,

TTT ′+=

′+= UUU (8)

Replacing these variables on the averaged equations, we obtain,

( ) 0=⋅∇ Uρ (9)

( ) ( )( ) fUUUUU ρρμρ +′′−∇+∇⋅∇+−∇=⋅∇T

pU (10)

( ) ( )TT hTh ′′−∇∇=∇ UU ρλρ (11)

Appearing as turbulent diffusive fluxes, the products of fluctuations impact the flow by increasing the

diffusivity of the variables. The physical interpretation is that the turbulent fluxes, UU ′′ρ and Th′′Uρ from advective origin, improves the mixing of momentum and energy. As previously mentioned above, the governing equations represent the steady state of the flow. In order to perform transient simulations, the

transient term has to be included, resulting in the so called Unsteady Reynolds Averaged Navier-Stokes Equations (U-RANS).

Within this approach, there are various ways to model the products of fluctuations. Among the most common models for industrial applications, we can find k-ε, k-ω and Reynolds Stress (RST) models. The two first approaches are included in the so called turbulent viscosity models. Basically these models solve new transport equations in order to estimate the production and dissipation of turbulence, and, from these variables, a turbulent viscosity is calculated, giving the increasing in the dissipation rate due to turbulent fluctuations. On the other hand, the Reynolds Stress model solves one transport equation for each component of turbulent flux, ρU’U’. The description of the various turbulence models are beyond the scope of this work and can be found in (Wilcox (1998), Hinze (1975), among others).

According to Silveira Neto et al. (2003), the use in unsteady simulations of the averaged equations is incompatible with the time averaging process, and so for the Unsteady Reynolds Averaged Navier-Stokes Equations – URANS, the averaging process should be interpreted as a filtering process, where the low frequency fluctuations are calculated and the high frequency fluctuations are modeled.

A more rigorous filtering process was proposed by Smagorinsky (1963). The purpose of this approach is to separate the turbulence scales in two parts, the large scales, which are calculated, and the small scales which are modeled. In order to do this, a suitable filter function, G(x,t) and a spatial and temporal characteristic scales xC and tC, have to be defined. These scales will determine respectively the cut wave number and frequency of the turbulent spectrum. For a generic variable φ, representing velocity component or temperature, the spatial and temporal filtered functions are,

( ) ( ) ( )∫ −=V

CC dGtt xxxxxx ,,, φφ

( ) ( ) ( )∫ −=T

CC dtttGttt ,,, xx φφ (12)

In numerical calculations, the spatial and temporal filtering process is made “naturally” by the spatial and temporal discretization process, i.e., the scales greater than the grid scale are solved and the lower are modeled. The same is valid for the time scales. Figure 2 shows a scheme of a typical turbulent spectrum and the cut frequency, given by the grid scales, where kC = 1/xC is the cut wave number. The turbulent fluctuations with wave numbers smaller than kC are solved and the fluctuations with higher wave numbers are modeled. The cut off wave number is inversely proportional to the grid scale given by, Δ = V1/3, where V is the volume of finite volumes cell so, the finer the grid, the higher the frequencies captured in computations.

Figure 2 – Scheme of a typical turbulent spectrum

Applying these filters to the instantaneous equations, we obtain,

( ) ( ) 0=⋅∇+∂∂ Ut

ρρ (13)

Log (k)

Log(

E K)

Log (kC)

( ) ( )fUUUUUUUUUU

UUU

ρρρρρμ

ρρ

+⎟⎠⎞

⎜⎝⎛ −−−′′−⎟

⎠⎞⎜

⎝⎛ ∇+∇⋅∇

+−∇=⋅∇+∂∂

''T

pt

(14)

( ) ( ) ⎟⎠⎞⎜

⎝⎛ −−−′′−∇∇=∇+

∂∂ UUUUU TTTTTT hhhhThht

ρρρρλρρ ' (15)

One can observe the existence of additional terms on these equations, when compared with the averaged equations. The averaging process is a particular case of the filtering process, when temporal and spatial scales are infinite. As an advantage of taking non-infinite scales is that the transient term is conserved in the equations for the non-filtered timescales, which is a more “natural” way to treat transient flows. The additional terms appearing in the filtered equations could be modeled in a similar way as is the Reynolds Stress Tensor. Nevertheless, past experience shows that the consideration of these additional terms does not significantly improve the results. Germano (1986) has proposed a more general way to decompose the advective term making,

( ) ⎟⎠⎞⎜

⎝⎛ −−−−′′=−′′−= UUUUUUUUUUUUUUτ ''ρρ

( ) ⎟⎠⎞⎜

⎝⎛ −−−−′′=−′′−= UUUUUUUq TTTTTTT hhhhhhh 'ρρ

(16)

and introducing these terms into the filtered equations. Then, the filtered equations are rewritten as,

( ) ( )( )( ) fUU

UUU

τ ρμ

ρρ

+−∇+∇⋅∇

+−∇=⋅∇+∂∂

T

pt (17)

( ) ( ) ( )qU −∇∇=∇+∂∂ Thht TT λρρ (18)

These equations are similar to the classic URANS equations but were obtained without any hypothesis about the timescales. At this point, the main difference between URANS and LES approaches is how the turbulent stresses and fluxes are modeled. As stated above, within the URANS approach, there are two ways to model the turbulent stresses, one is to solve additional equations in order to calculate a suitable turbulent viscosity which represents the additional dissipation due to turbulence. The other and more sophisticated approach, is to solve transport equation for each component of the Reynolds stress tensor,

UU ′′ρ . The decision between models is made based on the complexity of flow and the computational resource available. Within the LES approach, it is supposed that the modeled part of the turbulent spectrum is homogeneous and geometry independent, and so simple relations for the turbulent viscosity could be used. The most common relation for this viscosity is the one proposed by Smagorinski (1963) as,

( ) ⎟⎠⎞⎜

⎝⎛ ∇+∇=Δ=

T

SCSGS UUSSS21,22ν (19)

where CS is the Smagorinski constant and Δ is a grid subscale depending on the grid size. This relation shows the consistency of this approach, in the sense that the turbulent viscosity tends to zero when grid scales tends to zero. This shows that when grid and time steps are refined beyond the minor turbulent scales (Kolmogorov dissipative scales), this model approaches to DNS (Direct Navier-Stokes) calculations. On the other hand, URANS approaches could introduce artificial dissipation, even when refined grids are used, not restoring the instantaneous equations for this limiting case.

The theoretical values of the Smagorinski constant, based on isotropic turbulence in the inertial range spectrum, is CS = 0.18, but values from 0.06 to 0.25 are commonly used. In this work, the value of 0.25 was

used in order to stabilize numerical calculations for the relatively coarse grid used in large flow Re numbers.

In this work we have chosen the classic k-ε model to model the flow within the flue-gas valves and to compare results with those obtained using the LES approach.

The CFD Model A transient, 3D flow model was implemented for the flue-gas valves. As already mentioned, initially the k-ε approach was used to model the turbulence phenomena. The time step was selected in order to capture the frequencies with the same order of magnitude of those measured at the external tube wall (~250 Hz). Therefore, the time step selected in this case was 1.0e-3 seconds. For the LES approach, because of numerical instability problems, the time step had to be reduced to 1.0e-4.

All the simulations presented in this paper were performed using the commercial code CFX-5.6 (Ansys Inc. (2003)). This code applies the Element Based Finite Volume Method (Schneider and Raw (1987) and Maliska (2004)) for unstructured grids with a fully implicit coupled solver. This coupling includes mass and momentum equations and phase coupling in the case of multi-phase flows, i.e., solving the whole system of equations for both phases simultaneously.

The computational grid used was about 1.1 x 106 nodes for both approaches. At this point, it is important to explain that, for the LES approach, it is recommended in literature to simulate at least half of the logarithmic scale of the turbulent spectrum, i.e., the grid refinement should be such that log(kC) is in the mid of the total turbulent spectrum. The relation between the largest turbulent scale and the minor ones (Kolmogorov scale) is proportional to Re3/ 4. Considering that the major structures have the same order of magnitude of the valves opening (~1 m) and the working Re number has the order of magnitude of 5x106, the Kolmogorov scale would be of order of 1x10-5 m. Thus, the “ideal grid scales” should have the order of magnitude of 1 cm approximately. If we take this back to the domain scales, an ideal computational mesh with approximately 120x106 volumes would be necessary. This is obviously impracticable with computers available today; as such grid should be also resolved for a huge amount of time steps. Nevertheless, results obtained with the “coarse” grid used in this work, show that the methodology can provide the required information.

Figure 3 shows the computational domain used in the simulations. The elbow part upstream the valves were included in order to take into consideration the asymmetric inlet flow conditions.

Detail of Sliding Gate Valves

Figure 3 – Computational domain used in the simulations

Valve 1

Valve 2

Figure 4 shows the computational grid used to perform the simulations. ICEM CFD Version 4.2 was used for grid generation. Note that the cross section of the valves is rectangular, and the flow is more restricted in one direction, so results will be showed in vertical planes transversal and parallel to the valve sliding gates.

Figure 4 – Computational grid used in the simulations

Boundary conditions and fluid properties were chosen to represent the real operating conditions at the same time when acceleration measurements at external walls were performed. As it was stated, the final objective of the work is to perform a structural analysis, using fluid loads as inputs and compare the results with acceleration measurements at the tube wall.

A mass flow boundary condition was considered at domain inlet and a constant pressure condition at outlet. The outlet pressure value is a field measurement. After the second valve, there is a sequence of perforated plates called orifice chamber, which aims at reducing pressure and dissipating oscillations. As pressure value was available after the orifice chamber, this device was included in the simulation domain through a resistance offered by a porous domain proportional to the velocity.

Scalable wall functions were used for the k-ε model and the standard wall damping was used for the near wall treatment in the LES approach (see CFX 5.6 Manual for details).

CFD Results & Discussion Preliminary results were obtained with the k-ε model. Although some transient structures had been captured, its frequencies were around 10 to 20 Hz of magnitude, i.e., too low when compared to those frequencies obtained from the field measurements. Then, a model using the LES approach was implemented. Using this approach, pressure fluctuations with frequencies around 200 Hz has been captured. Results for both models are presented below. Initially, the global flow structure is presented using time averaged results in order to show the “mean” structure of the flow. Figure 5 shows the averaged velocities in vertical cut planes for the k-ε and LES models.

k-ε Model LES approach

Figure 5 – Time averaged velocity in vertical planes

Vector plots of averaged velocities helps to understand the flow pattern. Figure 6 shows vector plots of the averaged velocity after valves and, showing in the detail, the recirculation generated by the elbow preceding the first valve.

Valve 1 –k-ε model Valve 1 – LES approach

Valve 2 –k-ε model Valve 2 – LES approach

Figure 6 – Time averaged velocity vectors after valves

Overall, it could be seen that, for the LES approach, the jets generated after the valves penetrate more into the domain and have a significant less lateral diffusion. It is expected that the k-ε model to be more diffusive because it introduces an artificial diffusion in the model even for fine grids.

A picture of the instantaneous flow structure for both approaches could be seen in Figure 7 and Figure 8. Figure 7 shows snapshots of the vorticity contours in vertical planes and Figure 8 shows instantaneous velocity vectors after valves.

Plane YZ Plane XZ

Plane YZ Plane XZ

k-ε Model LES approach

Figure 7 – Snapshots of vorticity contours in vertical planes

It could be observed that, even using the same mesh, the structures captured within the LES approach are quite more complex, and many more local structures are captured. With the k-ε approach, the minor structures are dissipated because of the artificial diffusivity introduced by this model.

Valve 1 Valve 2

Valve 1 Valve 2

k-ε Model LES approach

Figure 8 – Instantaneous velocity vectors after valves

In order to analyze the dynamical flow behavior some numerical pressure sensors (monitors) were used at specified locations. As stated above, the U-RANS approach, using the k-ε model, did not predicted high frequency fluctuations, and, therefore, the results presented here, in term of transient behavior, will be focused on the LES approach. Figure 9 shows a typical pressure signal in a numerical sensor located 2 m downstream valve 1, at the midpoint of the duct. The FFT (Fast Fourier Transform) of the signal is also presented.

Monitor Point SLD1

122500

123000

123500

124000

124500

125000

125500

0.22 0.27 0.32 0.37 0.42 0.47

t [s]

P [P

a]

Figure 9 – Typical pressure spectra in monitoring point

Other monitors were used at different locations, but the signal was similar in all cases. Even when the pressure signal at the numerical sensors does not show the evidence of defined pressure patterns, the forces at walls obtained by integrating the pressure along them present peaks in clearly defined frequencies. This

Pressure Spectrum

0

50

100

150

200

250

0 50 100 150 200 250 300

ω [Hz]

P [P

a]

is a very impressive result. Figure 10 shows the x-component of the transient pressure forces force along the valves, and the respective FFT plot.

FORCE SLD1,X-DIR

-2000

-1000

0

1000

2000

3000

4000

0.2 0.25 0.3 0.35 0.4 0.45

t [s]

F [N

]

Force Spectrum

0

50

100

150

200

250

300

0 50 100 150 200 250 300 350 400

ω [Hz]

F [N

]

Figure 10 – Transient force on the valve 1 (x comp.)

Peaks at frequencies around 120 to 160 Hz and 220 Hz can be seen. Figure 11 shows the y-component of the force. In this case, the peaks are located at frequencies with the magnitude of 200 Hz.

FORCE SLD1,Y-DIR

-500

0

500

1000

1500

2000

2500

0.2 0.25 0.3 0.35 0.4 0.45

t [s]

F [N

]

Force Spectrum

-10

10

30

50

70

90

110

130

150

0 50 100 150 200 250 300 350 400

ω [Hz]

F [N

]

Figure 11 – Transient force on the valve 1 (y comp.)

The mean value of the forces is not zero due to the presence of the curves upstream the valve, and the asymmetrical opening of the sliding gates. For the second valve we have the similar situation but with peak frequencies between 120 to 150 Hz.

One preliminary conclusion obtained from these results is related to the type of input load which could make the structure respond as shown in Figure 1. Preliminary structural results, considering just the tube but not the valves (Medeiros et al. (2002)), showed that, when a distributed load is applied at the tube wall, the response for displacements did not present peaks at determined frequencies. On the other hand, when a localized load was applied in the same model, the response obtained presents peaks at very determined frequencies. Based on this information, there is an indication that forces on valves are transmitted to the tube through moments, generating peaks of displacements at the refractory failure points. This picture is schematically illustrated in Figure 12.

Figure 12 – Scheme of a possible situation causing the refractory break down at tube walls

This hypothesis is being verified in the structural analysis model where some characteristics are presented in the next section.

The Structural Model Preliminary results for the structural model were obtained using ANSYS 7.0. The transient forces originated by the LES approach were given as inputs for the model.

The flue gas duct structural modeling consists of the simulation of the external shell (of carbon steel plates) and internal cones (stainless steel plates), using SHELL 63 element, refractory lining simulated using SHELL 43 and the anchors simulated using BEAM 4. This combination was the fifth configuration which tried to simulate the flue gas duct and showed to be the best model to represent the dynamic behavior of this system. An additional benefit of this configuration is the possibility of disconnection of some of the anchors and/or the refractory lining fall down on certain region and its impact on the whole structure mechanical behavior.

The total model has 30464 elements and 20672 nodes. Figure 13 shows a section of the element distribution, while figure 14 shows a detail of the transition cone and refractory lining termination.

Fluctuating Force

Moment

Failure Point

Schematic wall deformation

Figure 13 – Flue gas duct and slide valve cone structural model

The force spectrum obtained from the CFD model was applied on the structural model at a point on each of the transition cones. A first modal analysis was performed and a transient dynamic analysis will be performed in sequence to be compared to the results obtained on the field readings obtained using accelerometers to validate the model.

Improvements will be implemented based on the validated dynamic simulation.

Figure 14 – Internal cone model detail

Conclusion A CFD model was implemented in order to compute transient fluid-dynamic forces to be used as inputs for the structural analysis using ANSYS.

The URANS approach, using k-ε model seemed not to provide the desired response. This is mainly due to the fact that this approach is based on a turbulent viscosity model and its characteristic is to dissipate the high frequency turbulent structures.

The LES approach, even using a “coarse” mesh, provide loads with peaks frequencies with the same order of magnitude to those measured in field operation. This appears to be a promising methodology to get turbulent loads on structures in order to perform fatigue analysis.

Results obtained using the LES approach show that this methodology could be very promising for the analysis of FSI (Fluid Structure Interaction) problems, especially the fatigue ones, i.e., those cases where the information of fluctuating quantities, not the average ones, are important for the structural analysis.

This work is one of the first applications of LES modeling used in a concrete industrial problem. One of the main goals of the work was to demonstrate the potentiality of this approach for some specific industrial zproblems, where the information of large turbulent scales are necessary.

This research is still underway, and the majority of the efforts are being introduced to improve the structural model built in ANSYS and to compare its results with the experimental ones. Another run with

Steel shell

Refractory lining

Refratory anchors

the LES approach is also being performed with a mesh approximately 3 to 4 times finer than the original one (about 4x106 nodes). This is being done to perform a grid influence study.

At the end of this research the authors expect to have a computational model that can predict the dynamic fluid and structural behavior of these flue-gas valves. This “computational tool” will help the Research and Development division of PETROBRAS to solve problems and make modifications to current installations where refractory is failing, generating undesirable stops for the whole unit.

References CFX 5.6 Users Manual Solver Theory Turbulence models, 2003, ANSYS Inc.

Germano, M., A proposal for a redefinition of the turbulent stresses in filtred Navier-Stokes equations, Phys. Fluids, vol. 29 (7), pp. 2323-2324, 1996.

Hinze, J. O., Turbulence, McGraw-Hill, New York, U.S.A., 1975.

Launder, B. E. and Spalding, D. B., The numerical computation of turbulent flows, Comp Meth Appl Mech Eng, 3:269-289, 1974.

Maliska, C. R., Computational Fluid Mechanics and Heat Transfer, LTC Editing, 2004 (in portuguese)

Medeiros, J.; Ronaldo, C. B.; Wendell, D. V., Análise Mecânico-Estrutural da Queda de Revestimentos Refratários utilizados em Dutos e Equipamentos de UFCC, PETROBRAS PROREC Program internal report, 2003 (in portuguese)

Schneider, G.E. and Raw, M.J., Control-Volume Finite Element Method for Heat Transfer and Fluid Flow Using Co-located Variables - 1. Computational Procedure, Numerical Heat Transfer, Vol. 11, pp. 363-390, 1987.

Silveira Neto, A.; Mansur, S. S. and Silvestrini, H. J., Equações da Turbulencia: Media versus Filtragem, Procedings of IX Brazilian Meeting of Thermal Engineering, Caxambu, MG, Brazil, (in portuguese)

Smagorinsky, J. General Circulation Experiments with Primitive Equations, Mon. Weather Rev., vol. 91(3), pp. 99-164, 1963

Wilcox, D, C., Turbulence Modeling for CFD, DCW industries, 1998.