Reynolds-averaged Navier-Stokes Framework Research Articles

Comparative Spray Atomization and Evaporation Characteristics of Dimethyl Ether and Mineral Diesel

Abstract Dimethyl ether is a new-generation alternative fuel to mitigate cold-start issues in compression ignition engines. It has a higher cetane number and can efficiently lead to superior atomization and evaporation characteristics. This computational study compares Dimethyl ether and baseline diesel sprays in a constant-volume spray chamber. This simulation study compares spray and evaporation characteristics en-hancement due to Dimethyl ether adaptation. Fuel properties greatly influence spray atomization and evaporation characteristics. This study is based on the Eu-lerian-Lagrangian approach adopted in the Reynolds-averaged Navier-Stokes framework. The liquid spray penetration obtained by simulations matched well with the experimental results of Dimethyl ether and baseline diesel. Spray model con-stants were tuned for diesel and Dimethyl ether separately, as the fuel properties of both test fuels are completely different. These tuned models were used to simulate Dimethyl ether and diesel sprays at fixed fuel injection timings and ambient condi-tions. Results showed a lower spray penetration length for Dimethyl ether than die-sel because of the flash boiling of Dimethyl ether. Smaller diameter droplets formed due to Dimethyl ether's lower viscosity, density, surface tension, and higher evapora-tion rate. The reduction in Sauter mean diameter was quite sharp after the start of injection for the Dimethyl ether. Diesel spray showed retarded atomization and evaporation compared to Dimethyl ether. The vapour penetration length of both fuels was almost the same; however, the vapor mass fraction was higher for Dime-thyl ether than baseline diesel. Dimethyl ether spray showed superior spray atomi-zation and improved evaporation of Dimethyl ether droplets.

Transonic Buffet Characteristics Under Conditions of Free and Forced Transition

Transonic buffet is commonly associated with self-sustained flow unsteadiness involving shock-wave/boundary-layer interaction over airfoils and wings. The phenomenon has been classified as either laminar or turbulent based on the state of the boundary layer immediately upstream of the shock foot, and distinct mechanisms for the two types have been suggested. The turbulent case is known to be associated with a global linear instability. Herein, large-eddy simulations are used for the first time to make direct comparisons of the two types by examining free- and forced-transition conditions. Corresponding simulations based on the Reynolds-averaged Navier–Stokes equations for the forced-transition case are also performed for comparison with the scale-resolving approach and for linking the findings with existing literature. Coherent flow features are scrutinized using both data-based spectral proper orthogonal decomposition of the time-marched results and operator-based global linear stability and resolvent analyses within the Reynolds-averaged Navier–Stokes framework. It is demonstrated that the essential dynamic features remain the same for the two buffet types (and for the two levels of the aerodynamic modeling hierarchy), suggesting that both types arise due to the same fundamental mechanism.

A comprehensive assessment of fractal wrinkling/eddy dissipation based combustion model for simulating conventional turbulent premixed and non-premixed flames

This paper presents a step-forward extension of the well-known chemistry/flow interaction Eddy Dissipation Concept approach for modelling both, non-premixed and premixed, combustion regimes in a Reynolds-Averaged Navier Stokes framework. The Eddy Dissipation Concept approach and its extended versions (e.g. Partially Stirred reactors, PaSR) are widely used for CFD modelling of a variety of combustion systems. This is mainly due to their capability of incorporating chemical kinetic rates in turbulent flows. However, in defining the averaged chemical reaction rates, EDC describes the fine-scale of turbulent reacting flowfield based on the so-called mixing-rate calculated from turbulent velocity field, which consequently makes this model more suitable for diffusion flame/combustion regimes. In a recent study by the present authors, a hybrid Wrinkling-EDC approach is introduced to model fine scales of turbulent reacting flows in MILD combustion regimes based on flame surface density in premixed flame regimes and turbulent intermittency in diffusion flame regimes. In the present study, we study the performance of this newly developed approach for the simulation of conventional turbulent flames. The simulations are performed and validated using well-documented turbulent premixed and predominantly non-premixed flames (e.g. Flame F3 of Chen et al. and Sandia Flame D). In addition, all cases are simulated using a recently developed EDC model (referred to as extended EDC) along with the standard EDC version where both their results are used as a reference. The obtained results reveal better performance of the Wrinkling-EDC approach over the standard and extended versions of EDC model for the simulation of turbulent premixed flame, and a comparable performance between the extended EDC and wrinkling-EDC approach for the predictions of species concentration and temperature fields of turbulent non-premixed flames.

Computational study of ECN Spray A and Spray D combustion at different ambient temperature conditions

The combustion process of the Engine Combustion Network (ECN) Spray A and Spray D is studied over a wide range of ambient temperatures from 750 K to 900 K. With nominal diameters of 89.4 μm for the Spray A and 190.3 μm for the Spray D, these n-dodecane sprays are representative for light- and heavy-duty compression ignition engine applications, respectively. Computational Fluid Dynamics calculations are carried out using a Lagrangian parcel Eulerian fluid approach in a Reynolds averaged Navier–Stokes framework. For the Spray D reference condition, two sub-grid flame structure assumptions and the effect of turbulence chemistry interaction (TCI) on autoignition and flame structure at quasi-steady state are assessed in the context of a well-mixed and an Unsteady Flamelet Progress Variable (UFPV) combustion model. After that, UFPV approach is used to evaluate combustion behavior for the different ambient temperature conditions. Reference condition results show that both well-mixed and flamelet assumptions lead to a similar autoignition sequence. In terms of ignition delay time, TCI plays an important role, within the UFPV model, in reproducing the experimental trend observed for the increase in nozzle diameter. In terms of lift-off length, the well-mixed model is observed to predict a longer value compared to the flamelet-based sub-grid assumption. Lastly, the analysis of the autoignition sequence and flame structure at quasi-steady state is also extended over the whole range of ambient temperature conditions.

The effect of high-pressure injection variations on the mixing state of underexpanded methane jets

This work presents a joint experimental and numerical study of global characteristics and mixing behavior of underexpanded methane jets at high-pressure conditions in a Constant Volume Chamber. Injection pressures of 200, 250, and 300 bar and pressure ratios of 4, 5, 6, 8, and 10 at each of those pressures have been investigated. Tracer LIF with acetone as tracer has been applied to experimentally quantify the mixing of methane and quiescent air. In order to exploit the symmetry of the configuration, accompanying simulations have been carried out in Reynolds-Averaged Navier-Stokes framework with the k – w SST turbulence model and real-gas modelling based on the Soave-Redlich-Kwong Equation of State has been employed to account for high-pressure corrections in thermodynamic and caloric properties. The experiments confirm the hyperbolic decay of axial fuel concentration and the Gaussian shape of traverse concentration profiles in the self-similar region of the jets, while simulation results match well with experimentally determined fuel concentration fields. It is found that scaling laws proposed in literature for steady-state jet propagation can qualitatively interpret the effect of injection variations on jet tip penetration and volume. Increasing pressure ratio at fixed injection pressure leads to the formation of slightly richer jets, with slightly smaller mass percentage in the range of air-to-fuel ratios most favorable to autoignition. By contrast, increasing pressure ratio at fixed chamber pressure leads to virtually identical Probability Distribution Functions of local air-to-fuel ratios and the same is observed when employing a fixed pressure ratio at higher pressure levels.

A population balance model for large eddy simulation of polydisperse droplet evolution

In the context of many applications of turbulent multi-phase flows, knowledge of the dispersed phase size distribution and its evolution is critical to predicting important macroscopic features. We develop a large eddy simulation (LES) model that can predict the turbulent transport and evolution of size distributions, for a specific subset of applications in which the dispersed phase can be assumed to consist of spherical droplets, and occurring at low volume fraction. We use a population dynamics model for polydisperse droplet distributions specifically adapted to a LES framework including a model for droplet breakup due to turbulence, neglecting coalescence consistent with the assumed small dispersed phase volume fractions. We model the number density fields using an Eulerian approach for each bin of the discretized droplet size distribution. Following earlier methods used in the Reynolds-averaged Navier–Stokes framework, the droplet breakup due to turbulent fluctuations is modelled by treating droplet–eddy collisions as in kinetic theory of gases. Existing models assume the scale of droplet–eddy collision to be in the inertial range of turbulence. In order to also model smaller droplets comparable to or smaller than the Kolmogorov scale we extend the breakup kernels using a structure function model that smoothly transitions from the inertial to the viscous range. The model includes a dimensionless coefficient that is fitted by comparing predictions in a one-dimensional version of the model with a laboratory experiment of oil droplet breakup below breaking waves. After initial comparisons of the one-dimensional model to measurements of oil droplets in an axisymmetric jet, it is then applied in a three-dimensional LES of a jet in cross-flow with large oil droplets of a single size being released at the source of the jet. We model the concentration fields using $N_{d}=15$ bins of discrete droplet sizes and solve scalar transport equations for each bin. The resulting droplet size distributions are compared with published experimental data, and good agreement for the relative size distribution is obtained. The LES results also enable us to quantify size distribution variability. We find that the probability distribution functions of key quantities such as the total surface area and the Sauter mean diameter of oil droplets are highly variable, some displaying strong non-Gaussian intermittent behaviour.

On the influence of wall distance and geometry for high-pressure n-dodecane spray flames in a constant-volume chamber

To isolate the effect of flame–wall interaction from representative operating conditions of an internal combustion engine, experiments were performed in a constant-volume pre-burn vessel. Three different wall geometries were studied at distances of 32.8, 38.2, and 46.2 mm from a single-hole 0.09-mm orifice diameter fuel injector. A flat wall provides a simplified case of flame–wall interaction. To mimic the division of a jet into two regions by the piston bowl rim in an engine, a two-dimensional confined wall is used. A third, axisymmetric confined wall geometry allows a second simplified comparison to numerical simulations in a Reynolds-averaged Navier–Stokes framework. As a limiting situation for a free jet, the distance from the injector orifice to the end wall of the chamber is 95 mm. Thermocouples installed in the end wall provided insights into local heat losses for reference cases without a wall insert. The test conditions were according to the Engine Combustion Network Spray A guidelines with an ambient temperature of 900 K and an ambient density of 22.8 kg/m3 with 15% O2. Flame structures were studied using high-speed OH* chemiluminescence with integrated single-shot OH PLIF and combined with pressure-based apparent heat release data to infer combustion progress and spray behavior. Soot was studied in a qualitative manner using high-speed natural luminosity imaging with integrated high-speed laser-induced incandescence. Overall, increased mixing upon interaction with the surfaces is observed to increase early heat release rate and to significantly reduce soot, with the nearest wall distance showing most effect. The flat wall gives rise to the most significant effects in all cases.

Dependence of convective boundary mixing on boundary properties and turbulence strength

Convective boundary mixing is one of the major uncertainties in stellar evolution. In order to study its dependence on boundary properties and turbulence strength in a controlled way, we computed a series of 3D hydrodynamical simulations of stellar convection during carbon burning with a varying boosting factor of the driving luminosity. Our 3D implicit large eddy simulations were computed with the PROMPI code. We performed a mean field analysis of the simulations within the Reynolds-averaged Navier-Stokes framework. Both the vertical RMS velocity within the convective region and the bulk Richardson number of the boundaries are found to scale with the driving luminosity as expected from theory. The positions of the convective boundaries were estimated through the composition profiles across them, and the strength of convective boundary mixing was determined by analysing the boundaries within the framework of the entrainment law. We find that the entrainment is approximately inversely proportional to the bulk Richardson number. Although the entrainment law does not encompass all the processes occurring at boundaries, our results support the use of the entrainment law to describe convective boundary mixing in 1D models, at least for the advanced phases. The next steps and challenges ahead are also discussed.

A Novel In Situ Flamelet Tabulation Methodology for the Representative Interactive Flamelet Model

ABSTRACTA priori flamelet tabulation methods have been used extensively to model premixed and non-premixed combustion in Reynolds-averaged Navier–Stokes (RANS) and large Eddy simulations (LES). Pre-tabulation of flamelet libraries have led to significant cost reduction of Computational Fluid Dynamics (CFD) simulations. However, these tabulation methods rely on certain simplifying assumptions. The dimensionality of the flamelet manifolds and size of flamelet libraries are also constrained due to memory and computational limitations. This limits the applicability and predictive capability of flamelet models. The traditional representative interactive flamelet (RIF) approach with multiple flamelets on the other hand involves online solution of flamelet equations and presumed probability density function (PDF) integration at each computational cell, making it attractive but computationally expensive and not suitable for LES with large cell counts. The current work discusses the development of a novel methodology that combines the advantages of an online flamelet solver along with the computational efficacy of tabulated models. The novel in situ tabulation approach consists of creating a flamelet table at each computational time step using the unsteady flamelet equations. The flamelets are then integrated in real time based on their respective scalar dissipation rate histories. This approach is capable of including history effects and unsteady chemical kinetic effects as it involves the online solution of unsteady flamelets. Implementation and comprehensive validation of the new framework is carried out against gas-jet flames as well as spray flames at high pressures. The approach is first validated for a partially premixed methane gas-jet flame in a RANS framework. The temperature and species mass fractions are compared against experimental results at different locations of a lifted flame. The approach is able to capture the transient flame characteristics. The model is then validated against an n-dodecane spray flame at high pressures in an LES framework with a 103-species n-dodecane chemistry mechanism. The model is able to capture ignition delays and flame liftoff of the n-dodecane spray over varying ambient temperature conditions. Due to the offsetting of presumed PDF integration cost, the proposed framework is shown to be faster than the traditional RIF approach, and the speedup over RIF increases with an increase in the cell count. This framework enables use of the RIF model for large cell count LES simulations.

3D hydrodynamic simulations of carbon burning in massive stars

We present the first detailed three-dimensional (3D) hydrodynamic implicit large eddy simulations of turbulent convection of carbon burning in massive stars. Simulations begin with radial profiles mapped from a carbon burning shell within a 15$\,\textrm{M}_\odot$ one-dimensional stellar evolution model. We consider models with $128^3$, $256^3$, $512^3$ and $1024^3$ zones. The turbulent flow properties of these carbon burning simulations are very similar to the oxygen burning case. We performed a mean field analysis of the kinetic energy budgets within the Reynolds-averaged Navier-Stokes framework. For the upper convective boundary region, we find that the numerical dissipation is insensitive to resolution for linear mesh resolutions above 512 grid points. For the stiffer, more stratified lower boundary, our highest resolution model still shows signs of decreasing sub-grid dissipation suggesting it is not yet numerically converged. We find that the widths of the upper and lower boundaries are roughly 30% and 10% of the local pressure scale heights, respectively. The shape of the boundaries is significantly different from those used in stellar evolution models. As in past oxygen-shell burning simulations, we observe entrainment at both boundaries in our carbon-shell burning simulations. In the large P\'eclet number regime found in the advanced phases, the entrainment rate is roughly inversely proportional to the bulk Richardson number, Ri$_{\rm B}$ ($\propto $Ri${\rm_B}^{-\alpha}$, $0.5\lesssim \alpha \lesssim 1.0$). We thus suggest the use of Ri$_{\rm B}$ as a means to take into account the results of 3D hydrodynamics simulations in new 1D prescriptions of convective boundary mixing.

Note on Turbulent Kinetic Energy Production for Reynolds-Averaged Navier–Stokes Models

Linear eddy-viscosity Reynolds-Averaged Navier–Stokes (RANS) turbulence models are based on the Boussinesq approximation that asserts the Reynolds stresses to be linearly dependent on the mean strain rate. Using the Boussinesq approximation for the Reynolds stress yields a production term in the turbulent kinetic energy equation that is proportional to the square of the magnitude of the strain rate tensor. For some flows, this relation to the strain causes overproduction of turbulence. Widely used ad hoc modifications of the production term using vorticity lead to an inconsistent energy balance in the mean flow kinetic energy equation, violating the energy conservation. In this note, how to obtain a consistent RANS framework for a given production term modification is shown.

Plasma-actuated Manipulation of Secondary Flow Towards Pressure Recovery Enhancement in a 3D Diffuser Modelled by an Eddy-resolving Second-moment Closure

A computational study is presented of the enhancement of pressure recovery in a three-dimensional diffuser by using plasma actuators, as experimentally investigated by Grundmann et al., Exper. Fluids 50, (1), 217–231 (2011). The flow configuration utilizes a streamwise-oriented arrangement of dielectric barrier discharge (DBD) plasma actuators mounted on the top wall of the inflow duct. Spanwise motion directed towards the duct corners was generated, modifying the inflow with locally intensified turbulence. The most advantageous enhancement was achieved using an actuator pulsed with 40 % duty cycle. For comparison an actuator operating continuously (100 % duty cycle) was also considered. The present computational study examines both cases, comparing them to the baseline configuration with no actuation. Simulations were performed within the scale-adaptive version of the Unsteady RANS (Reynolds-Averaged Navier Stokes) framework by using an eddy-resolving second-moment closure as the turbulence model. The plasma actuator was represented using a spatial body-force distribution based on PIV (Particle Image Velocimetry) measurements of the flow induced by a plasma actuator, Kriegseis et al., J. Phys. D: Appl. Phys. 48, 329401 (2015). The results obtained for both pressure-recovery enhancement and suppression correlate well with the experimental findings.

Comparisons of Horizontal-Axis Wind Turbine Wake Interaction Models

In this study, Reynolds-averaged Navier–Stokes (RANS) simulations are performed using the k-ε and k-ω shear stress transport (SST) turbulence closure schemes to investigate the interactions of horizontal-axis wind turbine (HAWT) models in the neutrally stratified atmospheric boundary layer (ABL). A comparative study of actuator disk, actuator line, and full rotor models of the National Renewable Energy Laboratory (NREL) 5 MW reference turbine is presented. The open-source computational fluid dynamics (CFD) code openfoam 2.1.0 and the commercial software ansysfluent 13.0 are used for simulations. Single turbine models are analyzed for turbulent structures and wake resolution in the downstream region. To investigate the influence of the incident wind field on very large turbine blades, a high-resolution full rotor simulation is carried out for a single turbine to determine blade pressure distributions. Finally, simulations are performed for two inline turbines spaced 5 diameters (5D) apart. The research presented in this study provides an intercomparison of three dominant HAWT models operating at rated conditions in a neutral ABL using an RANS framework. Furthermore, the pressure distributions of the highly resolved full rotor model (FRM) will be useful for future aeroelastic structural analysis of anisotropic composite blade materials.

Numerical Simulations of the Near-Field Region of Film Cooling Jets Under High Free Stream Turbulence: Application of RANS and Hybrid URANS/Large Eddy Simulation Models

This paper investigates the flow field and thermal characteristics in the near-field region of film cooling jets through numerical simulations using Reynolds-averaged Navier–Stokes (RANS) and hybrid unsteady RANS (URANS)/large eddy simulation (LES) models. Detailed simulations of flow and thermal fields of a single-row of film cooling cylindrical holes with 30 deg inline injection on a flat plate are obtained for low (M = 0.5) and high (M = 1.5) blowing ratios under high free stream turbulence (FST) (10%). The realizable k‐ε model is used within the RANS framework and a realizable k‐ε-based detached eddy simulation (DES) is used as a hybrid URANS/LES model. Both models are used together with the two-layer zonal model for near-wall simulations. Steady and time-averaged unsteady film cooling effectiveness obtained using these models are compared with available experimental data. It is shown that hybrid URANS/LES models (DES in the present paper) predict more mixing both in the wall-normal and spanwise directions compared to RANS models, while unsteady asymmetric vortical structures of the flow can also be captured. The turbulent heat flux components predicted by the DES model are higher than those obtained by the RANS simulations, resulting in enhanced turbulent heat transfer between the jet and mainstream, and consequently better predictions of the effectiveness. Nevertheless, there still exist some discrepancies between numerical results and experimental data. Furthermore, the unsteady physics of jet and crossflow interactions and the jet lift-off under high FST is studied using the present DES results.

Imbalance wall functions with density and material property variation effects applied to engine heat transfer computational fluid dynamics simulations

Heat transfer is a significant factor affecting internal combustion engine efficiency, emissions and performance. This study concentrates on model development for convective heat transfer and near-wall turbulent flow. The solution of complete fluid dynamics equations within the very thin-wall boundary layers is, and will be in the near future, very impractical in engineering scale flows with any present day method. An advanced numerical near-wall treatment method within the Reynolds averaged Navier–Stokes framework has been developed. The method solves simplified boundary layer equations for enthalpy, momentum, turbulent kinetic energy and dissipation in wall adjacent cells on cellwise high-resolution subgrids, adaptive to local conditions. The boundary layer equations include temperature gradient–induced density/multicomponent material property variations and complete imbalance contributions, for example, convection, transients, pressure gradient and external sources, in compact forms. The resulting numerical wall functions are valid with near-wall grid resolution ranging from the viscous sublayer to the fully turbulent region, thus avoiding the conflicting near wall resolution requirements of common low–Reynolds and high–Reynolds number turbulence models. The advanced near wall treatment method, comprising the numerical imbalance wall functions and accordingly modified low–Reynolds number turbulence model, is implemented in STAR-CD 4.12, extensively utilized in engine simulations. The near wall treatment method is validated against available measurements and direct numerical simulation data of strongly heated pipe flow. Performance of the near-wall treatment method in engine conjugate heat transfer simulations is also demonstrated. Local and average effects of variable properties and imbalance contributions on piston surface heat transfer, friction and turbulent sources are elaborated and contrasted to the standard high–Reynolds number near-wall treatment.

Computational modelling of the HyperVapotron cooling technique

Efficient heat transfer technologies are essential for magnetically confined fusion reactors; this applies to both the current generation of experimental reactors as well as future power plants. A number of High Heat Flux devices have therefore been developed specifically for this application. One of the most promising candidates is the HyperVapotron, a water cooled device which relies on internal fins and boiling heat transfer to maximise the heat transfer capability.Over the past 30years, numerous variations of the HyperVapotron have been built and tested at fusion research centres around the globe resulting in devices that can now sustain heat fluxes in the region of 20–30MW/m2 in steady state. Until recently, there had been few attempts to model or understand the internal heat transfer mechanisms responsible for this exceptional performance with the result that design improvements have been traditionally sought experimentally which is both inefficient and costly.This paper presents the successful attempt to develop an engineering model of the HyperVapotron device using customisation of commercial Computational Fluid Dynamics software. To establish the most appropriate modelling choices, in-depth studies were performed examining the turbulence models (within the Reynolds Averaged Navier Stokes framework), near wall methods, grid resolution and boiling submodels.Comparing the CFD solutions with HyperVapotron experimental data suggests that a RANS-based, multiphase model is indeed capable of predicting performance over a wide range of geometries and boundary conditions if suitable sub-models are developed. Whilst a definitive set of design improvements is not defined here, it is expected that the methodologies and tools developed will enable designers of future High Heat Flux devices to perform significant virtual prototyping before embarking on the more costly build and test programmes.

Comparative Validation Study on Identification of Premixed Flame Transfer Function

The flame transfer function (FTF) of a premixed swirl burner was identified from a time series generated with computational fluid dynamics simulations of compressible, turbulent, reacting flow at nonadiabatic conditions. Results were validated against experimental data. For large eddy simulation (LES), the dynamically thickened flame combustion model with one step kinetics was used. For unsteady simulation in a Reynolds-averaged Navier–Stokes framework (URANS), the Turbulent Flame Closure model was employed. The FTF identified from LES shows quantitative agreement with experiment for amplitude and phase, especially for frequencies below 200 Hz. At higher frequencies, the gain of the FTF is underpredicted. URANS results show good qualitative agreement, capturing the main features of the flame response. However, the maximum amplitude and the phase lag of the FTF are underpredicted. Using a low-order network model of the test rig, the impact of the discrepancies in predicted FTFs on frequencies and growth rates of the lowest order eigenmodes were assessed. Small differences in predicted FTFs were found to have a significant impact on stability limits. Stability behavior in agreement with experimental data was achieved only with the LES-based flame transfer function.

Three-Dimensional Flow Prediction and Improvement of Holes Arrangement of a Film-Cooled Turbine Blade Using a Feature-Based Jet Model

A feature-based jet model has been proposed for use in three-dimensional (3D) computational fluid dynamics (CFD) prediction of turbine blade film cooling. The goal of the model is to be able to perform computationally efficient flow prediction and optimization of film-cooled turbine blades. The model reproduces in the near-hole region the macroflow features of a coolant jet within a Reynolds-averaged Navier-Stokes framework. Numerical predictions of the 3D flow through a linear transonic film-cooled turbine cascade are carried out with the model, with a low computational overhead. Different cooling holes arrangements are computed, and the prediction accuracy is evaluated versus experimental data. It is shown that the present model provides a reasonably good prediction of the adiabatic film-cooling effectiveness and Nusselt number around the blade. A numerical analysis of the interaction of coolant jets issuing from different rows of holes on the blade pressure side is carried out. It is shown that the upward radial migration of the flow due to the passage secondary flow structure has an impact on the spreading of the coolant and the film-cooling effectiveness on the blade surface. Based on this result, a new arrangement of the cooling holes for the present case is proposed that leads to a better spanwise covering of the coolant on the blade pressure side surface.