ASME-JSME 2007 Thermal Engineering and Summer Heat Transfer Conference July, 08-12, 2007 - Vancouver, Canada
AJ-1266
DEVELOPMENT AND VALIDATION OF A C++ OBJECT ORIENTED CFD CODE FOR
HEAT TRANSFER ANALYSIS
L. Mangani, C. Bianchini, A. Andreini, B. Facchini
Department of Energy Engineering ”Sergio Stecco” University of Florence - Italy
Via Santa Marta, 3 - 50139 Florence - Italy [email protected]
ABSTRACT
This paper describes the development and validation steps of computational sub-models for gas turbine heat transfer applica-tions, within an open source CFD code based on the Field Opera-tion and ManipulaOpera-tion C++ class library for continuum mechan-ics (OpenFOAM, http://www.opencfd.co.uk). Open FOAM is based on a polyhedral finite-volume approach with a co-located variables arrangement. In order to set up OpenFOAM toolbox to analyze heat transfer problems with RANS approach, it was necessary to add and implement some additional sub-models. First of all a SIMPLE like algorithm was specifically developed to solve the fully three dimensional, steady state form of compressible Navier Stokes equations. Moreover several eddy viscosity models such as the standard, the Two Layer version and the realizable k − ε model and the k − ω SST model have been implemented. The accuracy of the implementations was vali-dated comparing results with experimental data available both from standard literature test cases and from in house performed experiments. The geometries considered as validation tests cover the typical heat transfer problems in gas turbine design . On the whole, during the tests, OpenFOAM code has shown a good ac-curacy and robustness. The purpose of this work is to show the ability of an innovative CFD tool as support for gas turbine de-signers and to verify its role as an effective substitute for standard commercial CFD packages.
INTRODUCTION
One of the most demanding problem in gas turbine design is the proper evaluation of heat transfer phenomena which in-volve all hot components of the engine. Furthermore, all topical design criteria make heat transfer problems more and more diffi-cult. An improvement in gas turbine performance, for example, can be produced by increasing turbine inlet temperature, which is usually well above the metal critical temperature. In addition, new design concepts adopted for combustors, based on lean pre-mixed flames, reduce the amount of air available for wall cooling. These are only two typical examples that justify the increasing interest in developing more and more advanced cooling systems. The complexity of geometries usually adopted in such designs and the high costs required for accurate heat transfer measure-ments justify the increasing use of CFD analysis in each phase of the design process. Nevertheless, CFD simulations for eval-uation of thermal loads and effectiveness of the cooling devices in gas turbine engines are demanding both in terms of physical modeling and geometrical mesh handling. Actual cooling ge-ometries are characterized, for example, by intricate shapes, with non aerodynamic turbulators such as pins and ribs that must be properly discretized, or they involve complex flows such as im-pinging jets that make turbulence modelling a key point. Such issues usually require CFD codes to satisfy some essential fea-tures: a quite large set of turbulence models, in order to have accurate predictions with all possible flows and the capability in handling hybrid unstructured meshes. The consequence of such strict requirements is that a very reduced set of CFD codes is available worldwide, and the choice is usually limited to few
well known commercial codes. Commercial software have dom-inated, in the last decade, CFD analysis of heat transfer problems for turbomachinery applications both in industrial and academic field. Besides their numerous advantages, such as the simplic-ity of use via practical graphical interfaces, they present some common drawbacks: for example the waste of system resources with a large part of packages not used in standard simulations, which is one of the source of their poor performances in terms of calculation times. However, according to experts, we think that the main drawback of commercial CFD codes is their nature of “black box solution maker”. Advanced users in heat trans-fer applications need to understand the physics and sometimes the use of “ad hoc” models or modifications suitable for specific cases. User subroutine features provided by commercial pack-ages become quickly inadequate as the complexity of modifica-tions grows. Furthermore R&D department of big companies usually need to tune built-in models in order to feed calculations tools with their design practice frequently based on detailed and expensive experimental tests.
The objective of the work presented in this paper is to show the capabilities of a new open-source software environment where it is possible to implement new models, renew the exist-ing ones and experiment with model combinations. The Open-FOAM package (Field Operation And Manipulation) [1, 2] is an object-oriented numerical simulation toolkit written in C++ lan-guage [3, 4, 5]. Besides its advanced basic native CFD features, which will be described in the next parts, its essential charac-teristic is the opportunity to build new models and solvers with high simplicity and in less time than with standard Fortran based codes. Object-oriented programming of C++ drastically reduces the probability of bugs introduction with a consequent saving in debugging time. This paper describes the attempt to build a CFD package suitable for typical steady state heat transfer analysis which could be able to assist gas turbine design process. As will be described later, to reach such goal it was necessary to intro-duce specific modules to the standard release in order to over-come the limitations of built-in approaches: first of all a com-pressible steady state solver capable at handling transonic flows, then a set of turbulence two-equations closures with particular reference to a detailed near wall treatment. Additional features such as temperature dependent thermo-physical properties and generic grid interfacing have also been developed. It’s impor-tant t remark that such features are not available in the released version of the toolkit, as built models are mainly focused on un-steady weakly compressible flows. As a consequence it’s not possible to draw out specific comparisons between default and developed models.
As confirmation of the work done a set of validation test-cases were performed. In particular, in this paper, we will fo-cus our attention on the validation of the code with some com-plex configurations typical of heat transfer problems such as film cooling and impingement cooling. Both film and impingement
cases were analyzed with single and multi-hole configurations. In particular the well known Sinha experiment was considered for single hole film cooling test [6], while for multi-hole case an experiment performed in our Department was chosen [7]. Film cooling geometries here considered belong to full-coverage film cooling case also known as effusion cooling, a promising tech-nique used in combustor wall and turbine end-wall cooling. For single hole impingement tests, we referred to the classical ER-COFTAC C.25 test [8] while for multi-hole case again an exper-iment performed in our Department was considered [9]. Com-parison with experimental data are reported in terms of adiabatic effectiveness for film cooling tests and wall heat transfer coeffi-cient for impingement runs. Furthermore, in order to verify the accurate implementation of selected turbulence models, a simple flat plate tests was considered, while to show the robustness of the steady state solver developed results of classical validation tests are briefly commented.
NOMENCLATURE
U Vector velocity [m s−1]
D Hole diameter [m]
h Heat transfer coefficient [W m−2K−1]
k Turbulent kinetic energy [m2s−2]
L Height of the jet [m]
p Pressure [N m−2] p0 Pressure corrector [N m−2] ˙ q Heat flux [W m−2] Re Reynolds number T Temperature [K]
Ts Turbulent time scale [s]
Pk Turbulence production term
= µt “ ∂Ui ∂xj+ ∂Uj ∂xi−23δij∂Uk∂xk ” ∂Ui ∂xj−23ρk ∂Uj ∂xj [kg m −1s−3]
S Tensor strain = 0.5“∂Ui∂xj +∂Uj∂xi” [s−1]
X Streamwise direction [m]
Y Spanwise direction [m]
Cρ Compressibility termRT1 [kg J−1]
H Diffusive discretization term [kg−1s m3]
Greeks
α Angle between hole and crossflow
η Adiabatic effectiveness(T∞−Taw )(T∞−Tc)
< η > Spanwise averaged effectivenessPn(T∞−Taw )(T∞−Tc)
ω Turbulence frequency [s−1]
ε Turbulence dissipation [m2s−3]
µt,ef f Eddy viscosity [kg m−1s−1]
ρ Density [kg m−3] Subscripts 0 Uncooled plate aw Adiabatic wall ∞ Crossflow c Coolant w Wall OpenFOAM
The OpenFOAM (Field Operation And Manipulation) code [10, 11, 12] is an object-oriented numerical simulation toolkit written in C++ language. The toolkit implements operator-based implicit and explicit second and fourth-order Finite Volume (FV) discretization in three dimensional space. Efficiency of execution is achieved by the use of preconditioned Conjugate Gradient [13]
and Algebraic Multigrid solvers and the use of massively parallel computers in the domain decomposition mode.
Being primarily a C++ library ready to create executables, OpenFOAM uses object based programming language. It means programmers can use OpenFOAM native classes both to define their own classes or to build new applications, such as solvers or utilities, with ease of development. Object oriented program-ming allows data abstraction, object orientation, operator over-loading and generic programming. It means that it enables the construction of new types of data specific for the problem to be solved, the bundling of data and operations into hierarchi-cal classes preventing accidental corruptions, a natural syntax for user defined classes and it easily permits the code re-use for equivalent operations on different types.
OpenFOAM native grid engine can handle meshes of arbi-trary polyhedra bounded by arbiarbi-trary polygons, giving a large flexibility in mesh generation, see Fig. 1. Switching to Open-FOAM way of thinking, programmers must approach a field based philosophy more than a cell or face based one, each phys-ical quantity (no matter what dimension, rank or size) is repre-sented by a single object and treated as a field.
(a) (b)
Figure 1. Polyhedral mesh example.
Differential operators can be treated like finite volume cal-culus (fvc) or finite volume method (fvm) operators. The first approach performs explicit derivatives returning a field, the sec-ond one is an implicit derivation converting the expression into matrix coefficients. The idea standing behind is to think about partial differential equations in terms of a sum of single differ-ential operators that can be discretized separately with different discretization schemes. Differential operators such as gradient, divergence, laplacian and curl have been overloaded for the dif-ferent types of field, giving each of them the most suitable mean-ing.
Implementing different types of equations is now only a
mat-ter of combining in a different way the same set of basic differen-tial operator. Just to give an example of the capability of such a top-level code, let’s consider a standard equation like momentum conservation:
∂ρU
∂t + ∇ · (ρU U ) − ∇ (µ∇U ) = −∇p . (1)
It can be implemented in an astonishingly almost natural lan-guage which is ready to compile source C++ code:
solve ( fvm::ddt(rho, U) + fvm::div(phi, U) - fvm::laplacian(mu, U) == - fvc::grad(p) );
letting programmers concentrate their efforts more on the physics than on programming.
Another important feature allowed by object programming is the dimensional check, physical quantities objects are in fact constructed with a reference to their dimensions and so only valid dimensional operations can be performed avoiding errors and permitting once again an easier understanding.
Even if OpenFOAM can be used as a standard simulation package, its tools are in general too rough to well predict cases of industrial interests.
Its strength in fact is not really to be a ready-to-use code but stands in being open not only in terms of source code but, what’s more, in its inner structure and hierarchical design, giving the user the opportunity to fully extend its capability.
The subject of this work is the preparation of a set of mod-els capable of transforming OpenFOAM in a complete calcula-tion suite for heat transfer turbomachinery simulacalcula-tions. In these initial steps we have focused our efforts in standard steady state RANS approach considering that it still represents the most com-mon for CFD design process. Nevertheless, OpenFOAM code is fully able to handle unsteady calculations and it is already equipped with a LES module: the relevant computational and post-processing costs followed by an inevitable simplification of near wall treatment usually prevent the use of LES approach on actual geometries. To extend LES to cover industrial flows at high Reynolds numbers, new approaches (hybrid LES-RANS, DES, URANS) must be used: they are all based on a mix of LES and RANS and they require further development which will be the matter of future work.
In the next part we are going to discuss the development of the solver algorithm considered for our application and,
nonethe-less, the implementation of specific turbulence models for Low-Reynolds and High-Low-Reynolds simulations.
SOLVER
In turbomachinery and heat transfer applications, involved fluid flows may usually cover a wide range of Mach regimes. In particular, it usually happens that different Mach conditions si-multaneously arise in the same domain. Such situation makes the accurate solution of viscous flows governing equations a com-plex task.
Most widely used algorithms for compressible flows cal-culation use density as one of the main independent variables and pressure is determined via an equation of state. As there is very little or no change in density for low subsonic or nearly incompressible flows, these density-based methods fail in such regimes. Their application in cases of incompressible or low Mach number flows is questionable, since in that situation the density changes are so small that the pressure-density coupling becomes very weak.
To avoid this weakness another class of methods, proposed originally for viscous incompressible flows [14, 15, 16, 17] and later extended to compressible flows [18, 19, 20, 21, 22, 23, 24] use pressure as the main independent variable also with the con-cept of the ‘retarded density’ [25, 26, 27]. Such pressure-based approach is founded on the SIMPLE algorithm (Semi-IMplicit Pressure Linked Equations) [14]. In this method, continuity equation is converted into an equation for pressure corrector overturning the linkage between pressure and density to extend applicability range up to zero Mach number. The SIMPLE algo-rithm uses a segregated approach where the equations are solved in sequential steps letting to the iterative process the care of the non-linearity as well as the coupling between equations. To bet-ter visualize the cycle of SIMPLE algorithm a flow-chart of the pseudo code is reported in Fig. 2.
For each transport equation, a system of linear algebraic equations is obtained. These are solved cyclically applying the preconditioned conjugate gradient. Keeping the coefficients in the algebraic equations fixed, generally one to ten iterations are performed in the inner cycle. Typically for velocity, temperature and pressure correction equation, iterations are stopped when the sum of the absolute residuals over the whole solution domain has fallen about three orders of magnitude, or the prescribed maxi-mum number of inner iterations has been reached. The equation of state is used to update density after new solutions for tempera-ture and pressure are obtained. After one cycle of inner iterations has been performed for each variable, the coefficients of the al-gebraic equations are updated using the newest values of all vari-ables, outer iterations. In this way the non-linearity and coupling of equations is accounted for. Outer iterations are stopped when the sum of the absolute residuals for each variable decreases of prescribed orders of magnitude or when the normalized sums are
smaller than a prescribed small number.
Figure 2. Flow chart of SIMPLE algorithm.
In this work we considered a pressure-based finite volume solver using a co-located variable approach suitable for calculat-ing steady-state flows at all speeds. The development of this class of methods in contrast with the standard SIMPLE technique lies in a more precise derivation of the equation for the pressure cor-rector, allowing the possibility of treating at the same time low subsonic, almost incompressible, and high compressible flows. Such methods have been validated in many heat transfer prob-lem configurations.
The pressure correction equation, details can be found in [28, 29, 30, 31], in the compressible form says:
∇ · (CρU p
0
) − ∇ · (ρH(∇p0)) = −∇ · (ρU ). (2) The role of Eq.(2) in the SIMPLE cycle is to enforce mass
con-servation, it is in fact derived from a combination of momentum conservation and continuity equation. In order to solve Eq.(2), attention should be posed on the fact that the pressure correc-tion equacorrec-tion now assumes a convective-diffusive form instead of a purely diffusive behavior like the original incompressible for-mulation. While the other steady-state form transport equations have to be relaxed in order to characterize the inertial physics lost by the elimination of the time derivative, for the pressure correction equation this cannot be done. Usage of usual implicit relaxation techniques on pressure corrector, in fact, corrupt mass conservation on single iteration steps breaking the concept stand-ing behind SIMPLE algorithm. In subsonic cases, standard Neu-mann conditions at inlet velocity boundary, like in incompress-ible tests, determine ill-defined problems for Eq.(2). Care must be taken in handling pressure correction boundary condition in order to solve in a well-posed manner such an equation [32]. A combination of Dirichlet and Neumann type condition for the in-let has been tested.
TURBULENCE MODELS
The correct modeling of turbulent quantities is fundamental in conducting heat transfer simulations, because of the simulta-neous importance of well predicting both the near wall behavior and the complex structures of the main flow [33, 34]. Correct predictions of thermal quantities and gradients inside boundary layers are necessary to establish whether or not the cooling sys-tem is efficient. At the same time wall properties are very depen-dent on the development of the free stream flow.
Usage of wall function approach has to be avoided because of the unpredictability of boundary thermal gradient and the fail-ure in predicting transitional, Low Reynolds as well as adverse pressure gradient flows.
First step in modeling flows close to solid walls has been the implementation of several so called Low Reynolds k − ε models [35]. The idea standing behind such models is to damp turbulent viscosity near the wall through a damping function fµgoing to-wards zero as the distance from the wall is reducing. Constants multiplying source terms in the turbulent dissipation equation are in some cases also damped. The basic structure of the models is the same for all of them differing in the tuning of the damp-ing functions and some extra sources in dissipation equation, as shown below: ∂ρk ∂t + ∇ · (ρU k) − ∇ · (µef f∇k) = Pk− ρε , (3) ∂ρε ∂t + ∇ · (ρU ε) − ∇ · (µef f∇ε) = f1Pk ε k− f2ρ ε2 k . (4)
Of the many Low Reynolds k − ε models proposed in litera-ture in the course of years, the models by Lien and Leschziner
[25], Lien [36], Abe et al [37], Chien [38], Chen et al [39], Hwang and Lin [40] and Lam and Bremhorst [41] have been im-plemented.
It is known from literature that in high strain rate regions eddy viscosity models overpredict turbulent kinetic energy: this problem is sometimes referred to as “stagnation point anomaly” [42]. These higher values of k are due to an overestimate of production term Pk. To avoid such overprediction linear depen-dence between Pkand |S|2should be bounded in regions where
|S| grows. This is achieved with a time scale bound, derived
from a “realizability” constraint for Reynolds stress tensor to be definite positive: Ts= µt Cµρk = min à k ε, α √ 6Cµ|S| ! (5)
This limiter proposed by Durbin [43] has been inserted in all Low Reynolds models above presented as an option to be switched on or off by the user.
Then, in order to match good near wall predictions with suit-able modeling of flow structures far from the wall, Two Layer
k − ε models have been implemented. Such methods consist in
patching together a one equation model in the near wall layer and a two equation High Reynolds model in the outer layer [44]. Both Wolfestein and Norris&Reynolds closure formulas [36] have been tried without significant discrepancies in the re-sults.
Last model to be mentioned is the k − ω SST: it includes the modification of the standard k − ω to avoid sensitivity to quite arbitrary freestream values of ω [45, 46]. The basic idea is sim-ilar to Two Layer models: two different approaches are merged together to model the two different flow regions. The sublayer and logaritmic model is the standard k − ω, chosen because of its robustness, the absence of damping function and Dirichlet type boundary conditions. From the wake region and outside the boundary layer the standard k −ε, written in terms of ω, has been preferred due to its good compromise in predicting different kind of flows.
RESULTS
Generalities
All the cases to be presented, apart from the flat plate one, have been chosen because already tested and analyzed with com-mercial solvers by the authors, with some results already pub-lished, see for example [9, 47]. Grid sensitivity analysis have been performed when those runs were set up and it’s not repeated in this case, the various meshes however guarantee a first node
y+ ≤ 1. All fluid domains are discretized via hexahedral
ele-ments except Goldman test, total number of eleele-ments for each test is reported in Tab. 1.
Table 1. Grid sizes (thousands of elements)
GAMM test 1 10.0
GAMM test 2 13.5
Goldman test 7.2
Flat plate 17.6
ERCOFTAC C25 Axial-symmetric Impingement Jet 69
1-Hole Impingement Jet 387.0
5-Holes Impingement Jet 1705.0
Sinha test 177.0
6-Holes Effusion 2000.0
Due to the great number of implemented turbulence models, a shortcut has been used to name most of them: the acronyms presented in Tab. 2 will be widely used in substitution of authors’ full name.
Table 2. Acronyms for the various turbulence models.
k − ε Low Reynolds by Abe et al. AKN
k − ε Low Reynolds by Yoder and Georgiadis CH
k − ε Low Reynolds by Lien et al. CLL
k − ε Low Reynolds by Hwang and Lin HW
k − ε Low Reynolds by Lam and Bremhorst LB
k − ε Low Reynolds by Lien and Leschziner LW
k − ε Low Reynolds by Lien LNR
Realizability constraint correction Real
Two Layer TL
k − ω SST SST
The convective spatial discretization used is based on the Normal Variable Approach (NVA), and named in literature as Self Filtered Central Differencing (SFCD) scheme [10].
To check convergence the arrest criterium has been defined as single scalar normalized residual lower than 10−6. Normal-ization factor, N orm, was not changed from the released version and is defined as:
Φref = Φ ,
S = A · Φ , Sref = A · Φref ,
Norm =X(|S − Sref| + |Q − Sref|) .
If this condition was satisfied for all scalar but pressure rector calculations were stopped. Due to its nature, pressure cor-rection residual is of no interest and, moreover, its initial value is set to zero at every iteration. To verify whether or not pres-sure field is still varying, the maximum module of the prespres-sure corrector, at convergence exactly null, is imposed to be less than 10 [P a], remember an averaged relaxation factor for the pressure corrector is of the order of 10−2.
Solver Validation Tests
GAMM tests
The developed calculation procedure has been used to solve a variety of problems in heat transfer applications. Here the em-phasis is on the high compressible flows. The capability of the present method is demonstrated by computing inviscid flow in a channel with a bump on the lower wall named GAMM test. This test case has been used by various researchers to test their algo-rithms [18, 48]. Application of the method to two different types of inviscid flow, transonic and supersonic, are presented below.
The width of the channel is equal to the length of the bump, and the channel length is equal to three lengths of the jump. For transonic calculation, the thickness-to-chord ratio is 10% while for supersonic flow calculations it is 4%. In transonic and super-sonic regime at inlet is assumed that flow has uniform proper-ties and the upstream far field variable values (except pressure in transonic case) are specified while at the outlet all variable (ex-cept pressure in transonic case) are extrapolated. At the upper and the lower boundaries wall slip condition is prescribed.
First case with imposed inlet Mach number M ain= 0.675, gives the Mach number distributions along the walls and density gradient magnitude contour plot shown in Fig. 3(a) and Fig. 4(a). In the supersonic case, M ain = 1.65, the flow results super-sonic all along the bump: Mach number distributions and den-sity gradient magnitude contour plots are shown in Fig. 3(b) and Fig. 4(b). These results correspond to reference solutions from literature [18, 48].
Fig. 3(c) and Fig. 4(c) show the Mach number distribution and density gradient magnitude contour plot under the same con-dition of supersonic case but with two bumps. As can be seen by comparing Fig. 3(b), Fig. 3(c) and Fig. 4 the second bump does not influence the flow upstream indicating that the solution algo-rithm correctly reproduces the hyperbolic behavior of the flow.
Goldman test
As example of highly compressible subsonic, we have re-ported the simulation of a test based on the work of Goldman
et al.[49]. It is a 2-D turbulent analysis of a stator blade at the
mid-span; the details of the geometry and the mesh are shown in Fig. 5(a). The Reynolds number, based on the chord length of the
-1.5 -1 -0.5 0 0.5 1 1.5 x 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 Mach Number Lower-Wall Upper-Wall
(a) Mach profile in transonic flow.
-1.5 -1 -0.5 0 0.5 1 1.5 x 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 Mach Number Lower-Wall Upper-Wall
(b) Mach profile in supersonic flow.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 Mach Number Lower-Wall Upper-Wall
(c) Mach profile in supersonic flow with two-bump geometry.
Figure 3. Profile Mach number in upper and lower wall.
blade and the free-stream velocity, is 500000 and the inlet Mach number is approximately 0.2.
A comparison of the predictions for blade loading (defined as the ratio of static pressure to the inlet total pressure) with the experimental data is shown in Fig. 5(b).
Turbulence Models Validation Test Flat Plate
A simple test of a flow over a flat plate has been consid-ered the best choice to start the validation and selection process of the turbulence models. Because of the large amount of ac-cessible data, the ease and velocity of the test, corrections and tuning could be carried out quickly and accurately. It has been properly checked the near wall behavior of both turbulent kinetic energy and turbulent kinetic energy dissipation. Due to the rela-tive simplicity of the case, runs have been performed with a very fine grid: first node y+≈ 0.1. The flow field being modeled is
(a) Transonic flow.
(b) Supersonic flow.
(c) Supersonic flow with two-bump geometry. Figure 4. Density gradient contour plots.
that reported by Wieghardt [50] and later included in the 1968 AFOSR-IFP Stanford Conference [51]. Details about flow con-ditions are listed in Tab. 3.
Table 3. Flow conditions for flat plate test
Inlet temperature 294.4 K
Inlet Mach number 0.2
Pressure 101400 Pa
Turbulence kinetic energy - k 23.6 m2s−2
Dissipation - ε 3365 m2s−3
Comparison axial loc. - x 4.6870 m
Results are reported in terms of non dimensional k and ε plotted versus non dimensional wall distance at an axial location where flow is fully developed, Fig. 6.
Apart from CH [38] and HW [40], all models result to be in good agreement with turbulent kinetic energy experimental data for y+ ≥ 40. The tendency, excluded the above mentioned
(a) Geometry mesh. 0 0.005 0.01 0.01 0.02 0.03 0.03 0.04 0.04 x(m) 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 Padim FOAM exp
(b) Pressure ratio over experimental data. Figure 5. Stator Blade analysis.
y+ ' 10. LB estimation viceversa is higher than
experimen-tal registration. Level of approximation results in being quite uniform for the different models with CLL[39] and AKN [37] slightly better in overlapping free stream values for k+.
Also for turbulent dissipation, models well predict outer layer behavior: all models apart from LNR [36] basically coin-cide for y+≥ 40. The major disagreements are registered inside
the viscous layer. There is no agreement in fact in predicting the peak both in terms of positioning and values. Due to the dif-ferent boundary conditions imposed to the dissipation, near wall behavior is quite different for each model: LB fails in predicting the peak, AKN and HW results in having pretty high wall values for ε. The models that best suit experimental data reported by Patel [35] are LNR and CLL.
Of all the tested Low Reynolds models the CLL has been chosen as the most reliable one and used as the example for Low Reynolds models in the following cases.
Heat Transfer tests Impingement Cooling
Among various possible techniques to enhance heat transfer rate, impingement cooling certainly presents very high cooling efficency, thus it’s commonly found both in typical blade and
0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 k + y +
exp Patel AKN CH CLL HW LB LW LNR (a) k+profile 0 20 40 60 80 100 0,00 0,05 0,10 0,15 0,20 0,25 0,30 + y +
1/xy+ exp Patel AKN CH CLL HW LB LW LNR
(b) ε+profile
Figure 6. Turbulence quantities profiles.
combustor cooling systems, operating in a wide range of design conditions. From a numerical point of view, impinging jet flows present several interesting aspects allowing deep evaluation of turbulence models.
The problem of a 2-D normal impinging jet of air has been performed following a test case by ERCOFTAC. Then, the first row of an array of holes was simulated, with a comparison of the experimental results obtained during the European project LOPOCOTEP, with different turbulence models. Further simula-tions of the complete array were done with the selected models. To obtain the desired heat transfer results, runs simulation with an imposed heat flux on the impact wall has been performed.
ERCOFTAC C25 Axial-symmetric Impingement
An incompressible flow of a turbulent air jet impinging onto a flat plate was modeled [8]. The impact surface is heated and kept at constant heat flux. From the experiment, the Nusselt number distribution for various jet Reynolds numbers is known. Fig. 7 shows the geometry of the test case. The diameter of the pipe is D = 0.004 m. The inflow velocity and turbulence conditions were obtained from the development of a 50 D upstream extruded inlet hole. For validation purpose, a Reynolds number of 23.000 and a distance of L/D = 2 were chosen. The heated surface was modeled as a wall at constant heat flux ˙q = 200 W/m2 . All
other walls were treated as adiabatic walls. The far-field bound-aries are modeled as mixed inflow/outflow pressure boundbound-aries.
(a) total (b) particular
Figure 7. Entire geometry and particular of the grid around the stagna-tion point. 0 1 2 3 4 5 0 50 100 150 200 250 300 350 400 450 N u D r/D Exp CLL AKN CLLReal SST T L
Figure 8. Nusselt number distribution along radius.
Fig. 8 compares the predicted Nusselt number distributions of 5 turbulence models, namely AKN, SST, CLL, CLLReal and Two Layer , with the experimental profile.
The predictions of all two-equation models used in this val-idation case are in good agreement with the experimental data far from the stagnation point. As known in literature [42] in the area around the stagnation point Low Reynolds models without realizability constraint fail and dramatically overpredict the peak in the heat transfer coefficient (error of almost 200%).
At the same time Two Layer SST and realizable mod-els show about the same peak value. The local maximum at
r/D ≈ 2 is not seen by the Two Layer and is slightly predicted
by the SST. On the contrary, the CLLReal is well predicting such peak only shifting a bit towards higher values of the radius.
1-Hole Impingement cooling Both this and the
fol-lowing case, simulating typical design conditions for impinge-ment cooling of a gas turbine, has been performed on the set up of an experiment done at the Energy Engineering Department of the University of Florence for the European project LOPOCOTEP (LOw POllutant COmbustor TEchnical Programme). Coolant is injected from a plenum through a perforated plate and impacts over a flat plate at uniform heat flux. The holes on the plenum compose an array of 10 − 11 spanwise rows per 9 streamwise holes. This test is simulating the behavior of the first row while in the following one 3 − 2 rows (the array of holes is staggered) for a total of 5 jets are impinging. For further details refer to [52]. Main flow parameters and grid are reported in Tab. 4 and Fig. 9.
Table 4. Flow conditions for 1-hole impingement test
Inlet Temperature 308.2 K
Outlet Pressure 85101 P a
Inlet Turbulence level - Tu ≤ 0.5% %
Rej 7600
Inlet Velocity 0.28956 m/s
Wall Heat flux 3000 W/m2
Simulations have been validated in terms of heat transfer co-efficient calculated with respect to inlet static temperature almost coincident for such low Mach number with inlet total tempera-ture. Adiabatic simulations have been done too, in order to check whether this approximation could be done or not.
First thing to notice from Fig. 10 is that, contrarily to ER-COFTAC test, CLLReal model fails in well predicting heat trans-fer coefficient around the stagnation point. Moreover, due to the potential core that is not extinguished at the wall, two unphysical spurious peaks are predicted at X/D ≈ 1.
Figure 9. Impingement single hole grid. -3 -2 -1 0 1 2 3 4 5 0 100 200 300 400 500 600 700 H T C [ W m -2 K -1 ] X / D Exp TL kOmega SST ChenLienReal
Figure 10. Heat transfer coefficient on impinged wall along symmetry line.
Two Layer and SST result in being almost equivalent both for the peak level and the far from the stagnation point values, with the Two Layer predictions slightly lower everywhere on the impinged surface.
5-Holes Impingement cooling This case refer to the
same set of experiments of the previous test. For this multi-hole simulation the plenum as been schematized with a big plenum where the inlet mass flow is imposed. Computational boundary conditions follow exactly the previous 1-hole test.
For this geometry only the Two Layer and k−ω SST models have been tested against experimental results in terms of heat transfer coefficient, see Fig. 11 and Fig. 12.
Both experimental and numerical data are sampled onto the two different lines connecting symmetry planes and then merged together in the zone where a relative minimum is localized.
0 10 20 30 40 0 100 200 300 400 500 H T C [ W m -2 K -1 ] X / D experimental TL path1 TL path2 SST path1 SST path2
Figure 11. Heat transfer coefficient along center lines.
(a) Experimental
(b) Two Layer
(c) k-ω SST
Figure 12. Heat transfer coefficient[W m−2K−1]distribution on
im-pinged wall.
Even if obtained results are in good agreement with exper-imental data far from the stagnation point, it should be noticed that predictions for the peak value are quite different from mea-sured data. Higher discrepancies on the even peaks are probably due to errors in the experimental measurements [52]. Comparing the two models, Two Layer predicts peak values a 10% better of the SST giving basically identical results outside the stagnation points area. In any case, it should be considered that tempera-ture gradients are quite small. A better agreement is expected for higher values of wall heat flux.
Film and effusion cooling
Among the different techniques used in the cooling of hot parts in a gas turbine engine, the injection of cooling air in the main flow, producing a thin film of air that isolates the walls from the hot gases, is one of the most used. Because of the complex interaction between air and hot gases during mixing, many dif-ferent injection hole shapes and distribution have been studied and a great amount of research work is still on going [7]. In particular, most recent developments in drilling capabilities al-low the manufacturing of wide arrays of micro-holes (diameters below 1 mm), currently referred to as effusion cooling. Even if this technique does not produce a film wall protection as in stan-dard film cooling, its most important feature is the heat removed by the passage of coolant inside the holes (heat sink effect): the great number of holes and their high length/diameter ratio (with angles below 30◦) allows to heavily increase the overall cooling effectiveness [53]. Effusion cooling represents the base in the thermal design of modern aero-engine combustors and its use in the cooling of turbine endwalls is also investigated [7].
Even if film wall protection may not represent the main cool-ing effect in effusion technique, the prediction of mixcool-ing between coolant and cross flow and the corresponding assessment of adi-abatic effectiveness, still represent some of the most difficult task in CFD analysis [47]. Despite the well known deficiency of stan-dard eddy viscosity turbulence models in the accurate prediction of jet mixing in cross flow, essentially due to the isotropy as-sumption for turbulent stresses [54, 55, 56], both k − ε and k − ω models are still widely used in industrial CFD computations.
Therefore we will analyze in this part the accuracy of Open-FOAM code in the prediction of adiabatic effectiveness in ef-fusion cooling geometries, using the set of turbulence models selected for heat transfer analysis of this work. As introduced above, two test-cases will be studied: the well known single hole experiment by Sinha [6] and an experimental multi-hole geome-try aimed at turbine endwalls cooling [7].
Sinha test Experimental data and geometries are based
on tests made by Sinha et al.[6]; local and spanwise averaged effectiveness is compared with calculated values. The geometry is a flat plate with a single row of holes, while flow conditions are listed in table 5.
In the chosen geometry, hot gas flows over a flat plate, while coolant is injected through one row of holes; upstream of the injection channel there is a plenum. Fig. 13 shows the fluid do-main and the different boundary conditions imposed; in partic-ular, symmetry planes pass through hole axis and half spanwise pitch. On all inlet surfaces mass flow rate and static tempera-ture are imposed, symmetry conditions ensure zero gradient over boundaries in span direction.
Single row configuration was mainly considered in order to have a reference geometry to compare results for different
turbu-Table 5. Flow conditions for Sinha test
Cross flow temperature 300 K
Coolant temperature 153 K Pressure 105 P a Density ratio - DR 2.0 Blowing-rate - M 0.5 Momentum ratio - I 0.125 Turbulence level - Tu ≤ 0.2 %
Cross flow velocity 20 m/s
Rec 15700
Figure 13. Calculation domain and boundary conditions (Sinha et
al.[6]).
lence models and calculation meshes. The performances of the same five turbulence models as single hole impingement case were analyzed, namely AKN, CLL, CLLReal, SST, TL. The grid used is a structured grid, see Fig. 14 in which it is also reported a magnification of the zone around the hole.
Figure 14. Mesh details near walls.
In Fig. 16, laterally averaged effectiveness downstream of the hole is shown. Local lateral effectiveness at 1, 10 and 15 diameters downstream, is also shown in Fig. 15. A map of wall
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Y / D exp CLL AKN TL SST CLLReal (a) 1D 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 exp CLL AKN TL SST CLLReal Y / D (b) 10D 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Y / D exp CLL AKN TL SST CLLReal (c) 15D
Figure 15. Spanwise distribution of film cooling effectiveness at various sections.
effectiveness as well as distribution over the symmetry plane is reported in Fig. 17 and Fig. 18. There is a fairly good agreement
between numerical and experimental results for all models used.
0 5 10 15 20 25 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 X / D Exp CLL AKN TL SST CLLReal
(a) Laterally averaged
0 5 10 15 20 25 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 X / D Exp CLL AKN TL SST CLLReal (b) Center line
Figure 16. Comparison between laterally averaged and local center line film cooling effectiveness.
Attention must be paid on the results at one diameter test section for the CLLReal model. The baffle underlined at the end of the spanwise direction is a consequence of a poor develop-ing of the boundary layer predicted by the model. Low values of turbulent viscosity do not dissipate the horse-shoe vortex up-stream of the hole. A low pressure zone drives coolant gas com-ing from the plenum around the jet. As a consequence there is more spreading of the film in spanwise direction and the evidence is a local maximum for the effectiveness at X/D ≈ 1.2. On the other hand, centreline values show that SST model has a deeper penetration of the jet, see also Fig. 18, showing a local minimum in the effectiveness profile just downstream the hole.
It’s clear that numerical simulations predict a coherent jet Fig. 17, thus severally underestimating coolant lateral diffusion.
(a) CLLReal (b) CLL (c) SST (d) Two Layer (e) Color map
Figure 17. Effectiveness distribution over the wall.
(a) CLLReal (b) CLL
(c) SST (d) TL
Figure 18. Sinha - Temperature distribution on symmetry plane[K].
This is not evident at 1 D downstream, but the effect grows pro-ceeding in cross flow direction. This behavior is mostly due to an isotropic modeling of turbulence near the wall, see Simon, Jubran, Azzi and Lakehal [57, 54, 55, 56]. Similar results can be found also in Andreini et al. [47] with commercial solvers.
6-Holes effusion cooling The geometry of this case is
a six holes flat plate interposed in between a plenum and a chan-nel at lower pressure. To enhance numerical stability, the plenum has been gridded as six different smaller plena each one with the same inlet mass flow imposed to respect total experimental cool-ing air mass flow Fig. 19. A summary of flow conditions can be found in Tab. 6.
Table 6. Flow conditions for 6-holes effusion test
Cross flow temperature 323 K
Coolant temperature 298 K
Pressure 7.0 · 104 P a
Density ratio - DR 1.103
Blowing-rate - M 0.2
Figure 19. 6-holr effusion cooling case mesh.
Results are reported in terms of spanwise averaged adiabatic effectiveness, see Fig. 20. Together with experimental data, relative approach predictions using L’Ecuyer and Soechting cor-relation with Sellers superposition criterion have been reported [7]. First of all it can be noticed that Two Layer model strongly
0 10 20 30 40 50 60 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 X / D Experimental OF-TL OF-SST Ecuyer-Soechting
Figure 20. Spanwise averaged adiabatic effectiveness.
improves matching of both experimental and correlative data in comparison to k − ω SST. Two Layer is still in slightly over pre-diction for the peak values especially for even peaks, for the odd ones in fact peak values result in being on the same level of the
previous hole, meaning that rows interaction is very weak. This lack is due to the assumption of isotropic behavior of the turbu-lent viscosity. Such effect is even stronger for the k − ω SST. This can be seen also on the map of adiabatic effectiveness in Fig. 21.
(a) Two Layer
(b) SST
(c) experimental
Figure 21. Comparison between laterally averaged and local center line film cooling effectiveness.
Both models qualitative well predict the correlative decay of the spanwise effectiveness downstream the holes. By looking at the two-dimensional effectiveness map in Fig. 21 it’s evident the reason why SST turbulence model over predict previously plotted data. In fact the jet predicted by SST model presents a different shape a significant lateral diffusion as well as a more coherent structure along the flow. Both phenomena could be at-tributed to a higher turbulent viscosity. Moreover, the SST model predicts a thinner and larger film which is able to keep its cool-ing potential becool-ing less affected by the interaction with the main flow in the shear layer.
CONCLUSIONS
A numerical investigation was set up to validate an open source CFD code based on object oriented programming lan-guage. Many different tests were performed representing the state of the art for the cooling systems in turbomachinery appli-cations. Validation of a pressure correction algorithm and various turbulence models have been made by comparison with experi-mental data on typical heat transfer geometries. Massive parallel calculation have also been tested for the multirow configuration simulations both for impingement and effusion cases by the use of LAM/MPI library http://www.lam-mpi.org.
The combination of the new built-in OpenFOAM libraries is able to reproduce the flow conditions with good accuracy for all the geometries studied. Good agreement with experimental data and with the common commercial software has been reached for impingement and effusion cooling configurations.
Further investigations have to be made especially for ef-fusion cooling simulations. First of all, implementation of anisotropic turbulence models is needed in order to correct the lack of spanwise diffusion.
The used object oriented language give us a very flexible way for implementing new turbulence models, solver algorithms, boundary condition types and physical models.
Future work will be concentrated on expanding the capabil-ity of the code to simulate fluid-structure interaction, with main focus in conjugate heat transfer analysis.
ACKNOWLEDGMENT
Many thanks to Dr. Hrvoje Jasak of Assistant professor (do-cent), Faculty of Mechanical Engineering and Naval Architec-ture (FSB), University of Zagreb, Croatia.and to OpenCFD Lim-ited 2004-2007, Reading UK
References
[1] Openfoam user guide. Technical report, OpenCFD Lim-ited.
[2] Openfoam programmers guide. Technical report,
OpenCFD Limited.
[3] Stephen R. Davis. C++ for Dummies. 5th Edition. [4] Bjarne Stourstroup. The C++ programming language.
Ad-dison Wesley, 1997.
[5] Stanley and Lippmann. C++ Primer. Addison Wesley 3rd ed.
[6] A. K. Sinha, D. G. Bogard, and M. E. Crawford. Film-cooling effectiveness downstream of a single row of holes with variable density ratio. ASME Journal of
Turbomachin-ery, 113:442–449, 1991.
[7] L. Arcangeli, M. Surace, L. Tarchi, D. Coutandin, and S. Zecchi. Correlative analysis of effusion cooling systems.
ASME Turbo Expo, (GT2006-90405), 2006.
[8] D. Cooper, D. C. Jackson, B. E. Launder, and G. X. Liao. Impinging jet studies for turbulence model assessment. part i: Flow-field experiments. Int. J. Heat Mass Transfer, (36): 2675–2684, 1993.
[9] A. Andreini, E. Di Carmine, B. Facchini, and M. Surace. Combustor liner cooling: numerical analysis of imping-ment geometries. 2005.
[10] H. Jasak. Error Analysis and Estimation for the Finite
Vol-ume Method with Applications to Fluid Flows. PhD
the-sis, Imperial College of Science, Technology and Medicine, 1996.
[11] H. Jasak, H.G. Weller, and N. Nordin. In cylinder cfd sim-ulation using a c++ object-oriented toolkit. SAE Technical
papers, (2004-01-0110).
[12] F. Juretic. Error Analysis in Finite Volume. PhD thesis, Im-perial College of Science, Technology and Medicine, 2004. [13] J. R. Shewchuk. An introduction to the conjugate gradi-ent method without the agonizing pain. Technical report, School of Computer Science, Carnegie Mellon University -Pittsburgh, PA 15213, 1994.
[14] S. V. Patankar. Numerical Heat Transfer and Fluid Flow. Taylor & Francis, USA, 1980.
[15] W. Malasakera and H. K. Versteeg. Computational Fluid
Dynamics. Longman Scientific, England, 1995.
[16] M. Peric. Numerical methods for computing turbulent flows. Technical Report VKI LS 2004-06.
[17] J. H. Ferziger and M. Peric. Computational Methods for
Fluid Dynamics. Springer, Germany, 2002.
[18] M. Peric, Z. Lilek, and L. Demirdzic. A collocated finite volume method for predicting flows at all speed. Journal of
Numerical Methods in Fluids, 16:1029–1050, 1993.
[19] M. Darbandi and G. E. Schneider. Application of an all-speed flow algorithm to heat transfer problems. Numerical
heat transfer, 35:695–715, 1999.
[20] C. M. Rhie. Pressure based navier-stokes solver using the multigrid method. AIAA Journal, 27(8):1017–1018, 1989. [21] W. Shyy and M. E. Braaten. Applications of a
general-ized pressure correction algorith for flows in complicated geometries. Advances and Applications in Computational
Fluid Dynamics - ASME Winter Annual Meeting, pages
109–119, 1988.
[22] W. Shyy and M. E. Braaten. Adaptive grid computation for inviscid compressible flows using a pressure correction method. AIAA, ASME, SIAM, and APS, National Fluid
Dy-namics Congress, pages 112–120, 1988.
[23] K. C. Karki and S. V. Patankar. Pressure based calculation procedure for viscous flows at all speeds in arbitrary con-figurations. AIAA Journal, 27(9).
[24] J. Rincon and R. Elder. A high resolution pressure based method for compressible flow. Journal of Computation and
Physics, 3:217–231, 1997.
[25] F. S. Lien and M. A. Leschziner. A pressure-velocity so-lution strategy for compressible flow and its application to shock/boundary-layer interaction using second-moment turbulence closure. Journal of fluids engineering, 115:717– 725, 1993.
[26] E. S. Politis and K. C. Giannakoglou. A pressure-based algorithm for high speed turbomachinery flows.
Interna-tional journal for numerical methods in fluids, 25:63–80,
1996.
[27] J. J. McGuirk and G. Page. Shock capturing using a pressure-correction method. AIAA Journal, 28(10):1751– 1757, 1990.
[28] F. Moukalled and M. Darwish. Tvd schemes for unstruc-tured grids. International Journal of heat and mass
trans-fer, 46:599–611, 2003.
[29] F. Moukalled and M. Darwish. Normalized variable
and space formulation methodology for high-resolution schemes. Numerical heat transfer. Part B, fundamentals, 26:79–96, 1994.
[30] F. Moukalled and M. Darwish. A unified formulation of the segregated class of algorithms for fluid flow at all speeds.
Numerical heat transfer. Part B, fundamentals, 37:103–
139, 2000.
[31] F. Moukalled and M. Darwish. A robust pressure based algorithm for multiphase flow. International journal for
numerical methods in fluids, 41:1221–1251, 2003.
[32] I. Senocak and W. Shyy. A pressure based method fot turbu-lent cavitating flow computations. Journal of Computation
and Physics, 176:363–383, 2001.
[33] L. Davidson. An introduction to turbulence models. Tech-nical Report Publication 97/2, Chalmers University of Technology, 2003.
[34] J. Bredberg. On two equation eddy-viscosity models. Tech-nical Report Internal report 01/8, Chalmers University of Technology, 2001.
[35] V. C. Patel, W. Rodi, and G. Sheuerer. Turbulence mod-els for near wall and low reynolds number flows: a review.
AIAA Journal, 26:1308–1319, 1993.
[36] F. S. Lien. Computational modeling of 3-D flow in complex
ducts and passages. PhD thesis, University of Manchester,
Institute of science and technology, 1992.
[37] K. Abe, T. Kondoh, and Y. Nagano. A new turbulence mod-els for predicting fluid flow and heat transfer in separating and reattaching flows-i. flow field calculations.
Interna-tional Journal of Heat Mass Transfer, 37:139–151, 1994.
[38] D. A. Yoder and N. J. Georgiadis. Implementation and validation of the chien k − ε turbulence model in the wind navier-stokes code. Technical Report 209080, NASA, Glenn Research Center, Cleveland Ohio, 1999.
[39] F. S. Lien, W. L. Chen, and M. A. Leshziner. Low-reynolds-number eddy-viscosity modelling based on non-linear stress-strain/vorticity relations. Engineering
turbu-lence modelling and experiments, 3:91–100, 1996.
[40] C. B. Hwang and C. A. Lin. Improved low-reynolds-number k − ε model based on direct numerical simulation data. AIAA Journal, 36(1):38–43, 1998.
[41] C. K. G. Lam and K. Bremhorst. A modified form of the
k−ε model for predicting wall turbulence. Journal of fluids engineering, 103:456–460, 1981.
[42] P. A. Durbin. On the k − ε stagnation point anomaly.
Inter-national journal of heat and fluid flow, 17(1):89–90, 1996.
[43] G. Medic and P. A. Durbin. Towards improved prediction of heat transfer on turbine blades. Journal of turbomachinery, 124:187–192, 2002.
[44] D. Lakehal, G. S. Theodoris, and W. Rodi. Three-dimensional flow and heat tranfer calculations of film cool-ing at the leadcool-ing edge of a symmetrical turbine blade model. International journal of heat and fluid flow, 22: 113–122, 2001.
[45] F. R. Menter. Two equation eddy viscosity turbulence model for engineering applications. AIAA Journal, 32: 1598–1604, 1994.
[46] F. R. Menter. Zonal two equation k − ω turbulence models for aerodynamic flows. AIAA Paper, (93-2906), 1993. [47] A. Andreini, C. Carcasci, S. Gori, and M. Surace. Film
cooling system numerical design: adiabatic and conjugate analysis. ASME paper, (HT2005-72042), 2005.
[48] K.C. Karki. A calculation procedure for viscous flows at
all speeds in complex geometries. PhD thesis, University
of Minnesota, 1986.
[49] L. G. Goldman, , and K. L. McLallin. Cold-air annu-lar cascade investigation of aerodynamic performance of core-engine-cooled turbine vanes: I solid vane performance and facility description. Technical report, NASA Technical Memorandum, 1977.
[50] K. Wieghardt and W. Tillman. On the turbulent fric-tion layer for rising pressure. Technical Report TM-1314, NACA, 1951.
[51] D. E. Coles and E. A. Hirst. Computation of turbulent boundary layers. In AFOSRIFP Stanford Conference, vol-ume 2. Stanford University, 1969.
[52] B. Facchini and M. Surace. Impingement cooling for mod-ern combustors: experimental analysis of heat transfer and effectiveness. Experiments in Fluids, (40):601–611, 2006. [53] K. M. B. Gustafsson. Experimental studies of effusion
cool-ing. PhD thesis, Chalmers University of technology,
De-partment of Thermo and Fluid Dynamics, 2001.
[54] A. Azzi and D. Lakehal. Perspectives in modeling film cooling of turbine blades by transcending conventional two-equation turbulence models. Journal of
turbomachin-ery, (124):472–484, 2002.
[55] A. Azzi and B. A. Jubran. Numerical modeling of film cooling from short length steam-wise injection holes. Heat
and mass transfer, (39):345–353, 2003.
[56] D. Lakehal. Near-wall modeling of turbulent convective heat transport in film cooling of turbine blades with the aid of direct numerical simulation data. Journal of
turboma-chinery, (124):485–498, 2002.
[57] T. Simon. Film-cooling lateral diffusion and hole entry ef-fects. NASA Lewis, (1996 Coolant flow management work-shop), 1996.
