Probabilistic CFD computations of gas turbine vane under uncertain operational conditions

Accepted Manuscript Probabilistic CFD Computations of Gas Turbine Vane under Uncertain Operational Conditions Mohamad Sadeq Karimi, Saeed Salehi, Mehrdad Raisee, Patrick Hendrick, Ahmad Nourbakhsh PII: DOI: Reference:

S1359-4311(18)33869-9 https://doi.org/10.1016/j.applthermaleng.2018.11.072 ATE 12953

To appear in:

Applied Thermal Engineering

Received Date: Accepted Date:

26 June 2018 19 November 2018

Please cite this article as: M.S. Karimi, S. Salehi, M. Raisee, P. Hendrick, A. Nourbakhsh, Probabilistic CFD Computations of Gas Turbine Vane under Uncertain Operational Conditions, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/j.applthermaleng.2018.11.072

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Probabilistic CFD Computations of Gas Turbine Vane under Uncertain Operational Conditions Mohamad Sadeq Karimia,b , Saeed Salehia , Mehrdad Raiseea,∗, Patrick Hendrickb , Ahmad Nourbakhsha a Hydraulic

Machinery Research Institute, School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran b Aero-Thermo-Mechanics, Universit´ e Libre de Bruxelles, F. D. Roosevelt Avenue 50, 1050 Brussels, Belgium

Abstract The stochastic computations of a NASA gas turbine vane are conducted to investigate the effects of the operational uncertainties on the flow and heat transfer characteristics of the NASA C3X blade. The blade contains ten internal cooling channels to remove heat load. In order to minimize the analysis error the full conjugate heat transfer methodology has been employed to simulate the behavior of external hot gas flows, internal cooling air passages and the solid blade simultaneously. The v 2 − f turbulence model is used and it is shown the predicted results are in acceptable agreement with the available experimental data. Total pressure, total temperature, turbulence intensity, turbulent lengthscale of the inlet and the outlet static pressure are assumed to be stochastic with uniform probability distribution functions. The effects of these uncertainties on flow and thermal fields as well as the blade temperature distribution are studied. The polynomial chaos method with polynomials order p = 3 is used to quantify the effects of operational uncertainties. The non-deterministic CFD results are found to be in close agreement with the experimental data. Uncertainties specially in inlet total temperature and turbulent length-scale play key roles on the hydrodynamic and thermal fields around the airfoil also the turbine vane temperature distribution. Keywords: NASA C3X turbine vane, Conjugate heat transfer, Internal cooling, Stochastic condition, Uncertainty quantification, Polynomial chaos expansion

1. Introduction Recent developments in Computational Fluid Dynamic (CFD) methods have enabled researchers to carry out more accurate studies on fluid flow and heat ∗ Corresponding

author Email address: [email protected] (Mehrdad Raisee)

Preprint submitted to Journal of Applied Thermal Engineering

October 16, 2018

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transfer in complex geometries. These methods have been applied under the assumption of known inputs whilst in practice uncertainties are present in the working conditions, models constants, geometry etc. Neglecting the stochastic nature of these inputs may lead to erroneous output results. Turbomachinery flows are examples of complex industrial flows which are highly sensitive to their operating conditions as well as their geometry. Therefore, to conduct reliable and accurate analysis of such flows, it is of paramount importance to consider inevitable uncertainties of the system in the analysis. Generally, uncertainties are categorized into two main groups, namely epistemic and aleatory. The first group relates to randomness arising due to lack on knowledge of the physical phenomenon and may be reduced. However, the second group are mainly due to intrinsic randomness of material properties, boundary conditions and geometry and thus are not reducible. In order to achieve reliable results in engineering applications of CFD predictions, it is of essence to consider all sources of uncertainty of the model. Several methods have been developed by researches to perform Uncertainty Quantification (UQ). The most common method for this purpose is Monte Carlo (MC) which is straightforward and can be easily implemented. However, this methods is slow-convergent and usually requires a large number of realizations to achieve satisfactorily accurate results. To improve the convergence rate of UQ analysis, several efficient methods have been proposed, one of which is Polynomial Chaos Expansion (PCE). This method is based on spectral representation of stochastic system output [1, 2] and developed firstly as an intrusive method requiring amendments in the current available deterministic codes. Due to complexity and high computational cost of this process in CFD problems and the fact that CFD software are mostly closed source, the Non-Intrusive Polynomial Chaos (NIPC) expansion has been developed and investigated in several researches [3–7]. The NIPC uses only a limited number of properly selected samples by utilizing the deterministic solvers. The different UQ methods have been widely used in the literature to investigate the effects of uncertainties on the characteristics of different industrial applications such as turbomachines [4–14]. To cite an example, Petrone and Nicola [8] conducted a research to obtain aerodynamic and structural characterization of horizontal axis wind turbine performance under uncertainty using the quasi Monte Carlo Latin Hypercube Sampling method. In the field of hydraulic machineries, the first UQ analysis has been reported by Salehi et al. [4]. Using polynomial chaos expansion they showed that the geometrical and operational uncertainties have a great impact on the pump head while efficiency was more robust. In the following, stochastic simulation of turbomachines is reviewed focusing on researches related to heat transfer. Since gas turbines and compressors are extensively used in industry, a great deal of research work has been devoted for quantification of uncertainties in these turbomachines using various UQ methods [5, 6, 9, 10]. For instance, Pecnik et al.[10] investigated the effect of operating condition and model uncertainties on laminar-turbulent transition in transonic gas turbine compressor using Stochastic Collocation methods [15]. The UQ 2

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analysis of heat transfer and turbulent flow in pin-fin passages using NIPC was performed by Carnevale et al. [6]. They considered the Reynolds number as the only uncertain parameter in their work with Gaussian probability density function. In addition to operating conditions, geometrical uncertainties have also been studied by a number of researchers [7, 11–14]. For example, Montomoli et al. [11] studied the effects of variation of filleted edge on internal cooling of a gas turbine blade using Sampling method. Moreover, Dow and Wang [12] studied the impacts of compressor blade geometrical variation caused by manufacturing tolerances and in-service erosion on the performance of the system. In their research, Gaussian random field model of manufacturing uncertainty coupled with a probabilistic, gradient-based optimization frame work to optimize both geometrical configuration and tolerances simultaneously. In a recent research work on UQ analysis of gas turbine heat transfer, Mohammadi and Raisee [7] used full and sparse polynomial chaos methods to observe the effect of operational and geometrical uncertainties on the performance of an internal cooling system of gas turbine blade. Their research showed that despite the minor deviation of average Nusselt number, the local Nusselt number is highly affected by uncertainties. As gas turbine efficiency is strongly proportional to the turbine inlet temperature, a common method for achieving higher turbine efficiency is blade internal cooling which leads to increase in allowable turbine inlet temperature. One of the well known research work in this subject was conducted by Hylton et al. [16]. They measured the blade temperature, the local heat transfer coefficient and the static pressure on the blade midspan for two different internally cooled blades known as: “C3X and “MARK II. Their experimental results have been used for CFD code validation in a large number of investigations such as [17–22]. In [17], Bohn et al. performed a 2D analysis on a turbine guide vane cooled by air flowing through ten radially cooling channels by conjugate method. They verified their results against the experiments of Hylton et al. [16] for the blade “MARK II”. Using conjugate heat transfer method, Facchini et al. [20] simulate the flow around the C3X turbine vane. Three turbulence models namely; the low-Re k − ε, the RNG k − ε and the high-Re k − ε models were used and models predictions were compared with the measured data of Hylton et al. [16]. The predicted heat transfer coefficient and blade temperature deviated significantly from the experimental data, especially at the leading edge. The discrepancies could be due to inability of turbulence models in reproducing the boundary layer transition phenomenon or uncertainties in inlet and outlet conditions. Lou and Razinsky [21] performed numerical predictions for flow quantities as well as metal temperature distribution of NASA turbine blade C3X by utilizing three turbulence models namely; the k − ε, the nonlinear quadratic k − ε and the v 2 − f turbulence models. Numerical results of fluid flow simulation around the blade in subsonic and transonic conditions revealed the superior capability of the v 2 − f turbulence model in prediction of the boundary layer transition. Despite the above attempts, there are still remarkable differences between 3

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the numerical results and experimental data that could be originated from disregarding uncertainties. Although a great deal of studies have been conducted on gas turbine performance under uncertainties, no study is carried out on the impact of operational uncertainties on detailed flow and thermal fields considering conjugate heat transfer in a real gas turbine vane, whilst a more accurate route for the computation of the temperature field in gas turbine blade is full conjugate heat transfer analysis [23, 24]. Consequently, the prime motivation of the current work is to perform a thorough non-deterministic CFD study of NASA C3X gas turbine vane by taking into account co-current conduction and convection heat transfer in the solid and fluid domains of the blade. As all previous numerical researches on C3X turbine vane have been conducted under deterministic boundary conditions, the current research can be advantageous to achieve a realistic and reliable results that are well verified against the experiments. 2. Investigated Turbine Vane In the current study, the C3X turbine guide was selected to be studied. As mentioned in Introduction, this test case has been experimentally investigated by Hylton et al. [16] to obtain internal and external convection heat transfer coefficients besides metal temperature. The experimental setup comprised of linear cascade of three vanes where the middle one was air-cooled. As shown in Fig. 1, cooling process was carried out by passing fully-developed coolant air through ten radially channels expanding from hub to shroud. According to [16], the vane spacing and height are 117.7 mm and 76.2 mm, whilst the true and axial chords are respectively 144.9 mm and 78.2 mm as shown in Fig. 1. In Ref [16], principal independent operating conditions namely, Mach number, Reynolds number, turbulence intensity and wall-to-gas temperature ratio are varied for 18 different runs to assess the influence of each parameter. In the current study, a subsonic operating condition, i.e. run no. 112 with M = 0.90 and Re = 2.0 × 109 , is selected for computations. At the inlet, total pressure, total temperature and turbulence intensity are respectively 3.217 bar, 873 K and 8.3%. The coolant channels diameters and their flow rates were designed in such a way that the blade surface temperature remains approximately uniform. The diameters and mass flow rates of the coolant channels are given in the Table 1. 3. Deterministic Simulation

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In this section, the governing equations of flow and temperature fields, the utilized numerical approaches and the computational grid are introduced. 3.1. Mathematical Formulation The tonsorial form of density-averaged mass conservation, momentum and energy equations governing compressible turbulent flow in statistically station-

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Figure 1: The C3X vane and the channels locations. Table 1: Properties of the coolant channels

Channel No. 1 2 3 4 5 6 7 8 9 10

Diameter (mm) 6.30 6.30 6.30 6.30 6.30 6.30 6.30 3.10 3.10 1.98

Flow Rate (kg/s) 7.79 6.58 6.34 6.66 6.50 6.72 6.33 2.26 1.38 0.68

Tmid (K) 409.1 409.4 391.5 397.2 376.9 434.9 391.5 107.6 466.4 516.2

ary condition are expressed as follow [25]:

∂ ∂ p¯ ∂ (˜ ui ρ¯u ˜j ) = − + ∂xj ∂xi ∂xj

∂ (¯ ρu ˜j ) = 0, ∂xj

(1)

   ¯ ij ∂u ˜i ∂u ˜j 2 ∂u ˜k ∂R µ ˜ + − δij − , ∂xj ∂xi 3 ∂xk ∂xj

(2)

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  !  ∂  ∂ ∂u 2 ∂u ∂u i j k ˜ = u ˜j ρ¯H ui µ + − δij ∂xj ∂xj ∂xj ∂xi 3 ∂xk    ∂ ∂T ∂  g ˜ − −kT − ρ¯ uj H − u ˜j H , ∂xj ∂xj ∂xj

(3)

where the enthalpy is ˜ = H

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 ¯ ii u ˜i u ˜i R p¯ + + . 2 2¯ ρ ρ¯

(4)

It should be mentioned “ ˜ ” and “ ¯ ” respectively refer to density-weighted 00 u00 , ¯ ij = ρ¯u^ average and filtered part. The unknown Reynolds stress tensor, R i j is estimated by the adopted turbulence model. As reported in the literature [21], the v 2 − f turbulence model [26, 27] is able to reproduce the boundary layer transition the best in comparison to other turbulence models. Hence, in this study, the v 2 − f model was used. 3.2. Numerical Methods and Boundary Conditions

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The commercial software of ANSYS Fluent R18.2 was used to simulate the full conjugate heat transfer. In this solver, the governing equations are solved on a collocated grid via a fully-coupled element based finite volume method. The second-order upwind scheme was used for the discretization of the convective term in all transport equations and the convergence criteria was set 10−5 for all variable. As illustrated in Fig. 2, the computational domain consists of three stationary domains, namely; the hot gas flow field, the solid blade and the coolant channels. The 3D fluid domain is merely one blade extended 76.2 mm and located in the middle. In addition, periodic boundary conditions have been set for the domain to simulate the linear cascade configuration. Similar to the test condition reported in [16],the inlet and outlet of the fluid domain is located at 140 mm upstream and downstream of the blade leading edge respectively as shown in Fig. 2. At the hot fluid inlet, total pressure (p0,in ), total temperature (T0,in ), inlet turbulence intensity (Iin ) and inlet turbulent length-scale (lin ) were set as boundary conditions. The first three parameters were given in [16] but the turbulent length-scale was not reported.

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Figure 2: Computational domains including blade, hot gas flow field and cooling channels.

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In similar research studies on C3X gas turbine guide vane, different values for inlet turbulent length-scale were used. For example, in Lou and Razinski [21], lin was set to 0.5% of axial chord (i.e. 0.4 mm) while in Facchini et al. [20] it was assumed to be 5 mm. In this research, preliminary trial and errors were made to obtain the most appropriate value for this quantity. Fig. 3 displays the variation trend of Normalized Root-Mean-Square Error (NRMSE) of numerical results and experiments in [16] with respect to chosen inlet turbulent lengthscale. The results for two main parameters of blade heat transfer coefficient and temperature in midspan are presented. The outcome showed that by setting the inlet turbulent length-scale to 1% of the chord, the most accurate results can be achieved. At the outlet, the static pressure with respect to the reported Reynolds and Mach numbers was set as the boundary condition. The shroud and hub walls were set to adiabatic no-slip wall. Under the assumption of fullydeveloped flow at the cooling channel inlet and linear temperature distribution through the channel, the given coolant inlet temperature (i.e hub of the blade) agrees well with the reported midspan temperature [16]. Moreover, the inlet pressure is not reported in [16] and therefore its value is determined with reference to the given mass flow rate of each channel. Next, the inlet velocity profile of the coolant is obtained by considering the 1/7 power law velocity profile and the reported mass flow rate. The physical properties of air including thermal heat conductivity (kT ), constant pressure specific heat (Cp ) and dynamic viscosity (µ) were determined based on static temperature and fifth degree polynomials fits to the data reported by Rohsenow et al. [28]. The vane material is stainless steel of type AISI 310 with relatively low thermal conductivity [16]. The density and specific heat of the vane material are 7900 kg/m3 and 582.2 kJ/(kg.K) respectively. 7

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Figure 3: Normalized Root-Mean-Square Error of numerical results and experiments with respect to the inlet turbulent length-scale for blade heat transfer coefficient and temperature in midspan plane.

Figure 4: Generated grid including blade, hot gas flow field and cooling channels.

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3.3. Mesh generation and Grid study For the purpose of mesh generation, the domain is decomposed into three parts, namely blade, hot gas flow and coolant flow fields and each mesh is generated separately. The grid of blade and coolant flow fields are generated in Ansys-Meshing, while the mesh grid of hot gas flow field is produced using Ansys-Turbogrid. A fully structured gird is used for the coolant flow channels, while a quad pave mesh is produced for the blade and unstructured hot gas flow field core cells with several layers of boundary layer mesh on walls (see Fig. 4). Since C3X is an extruded profile with constant cross section, a 2D mesh including hot gas, solid domain and coolant fluid, was first generated. In the next step, the 2D mesh was extended 76.2 mm in the cross-stream direction to generate the whole domain. One of the most important steps in numerical simulation is assuring the independency of results with respect to the grid size. Therefore, in the current 8

research, five different mesh grids were generated. The details of these meshes are given in Table 2. All of these grids are clustered on the walls to resolve + the turbulent boundary layer, in a way that ymax at the blade surface decreases from 3.89 to 0.75 with an increase in mesh size from 0.9 × 106 to 4.0 × 106 . Table 2: Properties of different meshes

Mesh size (thousand cells) Total Blade Coolant Hot gas 904 1243 1715 3258 3967

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+ ymax at

Side walls 63.3 14.5 35.7 2.62 1.38

Blade 3.89 1.09 0.84 0.79 0.75

Fig. 5 displays the variation of average heat transfer coefficient of the blade ¯ blade surface ) and drag coefficient (CD = FD /( 1 ρU 2 Aref ), where outer surface (h 2 FD , ρ, U and Aref denote to drag force, fluid density, flow velocity and projected area of the blade respectively) with respect to the mesh size in logarithmic scale. All results are obtained at the design operating condition. It is seen that the mesh refinement is highly affects the calculated results. As can be seen, by mesh refinement from 0.9 × 106 to 4.0 × 106 , the drag coefficient and average heat transfer coefficient of the blade surface respectively change by about 26% and 4%. While increasing the mesh size to 1.7×106 influences the predictions, further mesh refinement has no sensible effect on the results. Since the low-Reynolds v 2 − f turbulence model is used for numerical simulation, it is necessary to + use a mesh grid which result in the ymax ≤ 5 for all walls. This ensures that at least one grid node is located with in the viscous sublayer [29]. Thus, the conjugated heat transfer computations of current study were obtained using a computational mesh with 3.3 × 106 nodes. 0.7

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¯ of the C3X turbine vane at Figure 5: Effect of mesh refinement on the predicted CD and h the design operating condition.

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4. Stochastic Conditions To obtain meaningful CFD results, defining the realistic operational conditions is crucial in numerical simulations. In the current work, numerical results were achieved under the assumption of stochastic operational conditions (i.e., inlet and outlet boundary conditions). The uncertainties are imposed on the measured parameters at the boundaries, namely, the inlet total pressure (p0,in ), the inlet total temperature (T0,in ), the inlet turbulence intensity (Iin ), the inlet turbulent length-scale (lin ) and finally the outlet static pressure (pout ). Consistent with similar researches [5, 9, 30, 31], these uncertain parameters are assumed to have a symmetry Beta probability distribution function (PDF) (Fig. 6): (1 − ξ)α (1 + ξ)β Γ(α + β + 2) Beta PDF(ξ) = , (5) 2α+β+1 Γ(α + 1)Γ(β + 1) with α = β = 4 and mean and standard deviation values identical to the measurement of Hylton et al. [16]. Figure 6 shows the PDF of β in the interval of [0,1]. 2.5

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As summarized in Table 3, the standard deviations (StD) for the pressure, temperature and turbulence parameters were respectively set to 0.007 (bar), 11 (K) and 1 % of their mean values. Table 3: Operational uncertainties.

Parameter Inlet total pressure, p0,in (bar) Inlet total temperature, T0,in (K) Inlet turbulence intensity, Iin (%) Inlet turbulent length-scale, lin (mm) Outlet static pressure, pout (bar)

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Mean (µ) 3.217 783 8.3 1.44 1.878

StD (σ) 0.007 11 0.83 0.144 0.007

Distribution Beta Beta Beta Beta Beta

5. Probabilistic framework

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5.1. Non-Intrusive Polynomial Chaos Expansion Consider y = U(ξ) as a physical or mathematical model where y and ξ = {ξ1 , ξ2 , · · · , ξd } ∈ Rd represent the model response or the quantity of interest (QoI) and the set of input variables respectively. The model response y is stochastic when the input vector ξ is uncertain through a joint probability density function f (ξ). By employing homogeneous chaos theory [2], Soize and Ghanem [32] showed that one can represent the model response as a series of orthogonal polynomial basis when input stochastic variables ξ are independent. Therefore, a random field of order p for d random variables ξ ≡ {ξi }di=1 can be represented in polynomial chaos form as follows: u(x; ξ) =

P X

ui (x)ψ i (ξ),

(6)

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where the number of terms in the summation is obtained from: P + 1 = p+d = d (p + d)!/(p!d!). In equation (6), ui ’s are unknown parameters which need to be identified and ψ i (ξ)’s belong to a set of multivariate polynomials which are orthonormal with respect to the joint probability density function f (ξ), i.e. hψ i (ξ), ψ j (ξ)i = δij , where δij represents the Kronecker delta [33]. Consequently, Jacobi polynomials are the corresponding basis of the PCE in the current study, since the distribution of all assumed uncertain parameters are considered to be Beta distribution. 5.1.1. Calculation of Statistical Moments In the current study, the regression method [34] is utilized to compute the unknown coefficients of the PCE. First, a set of N = 2(P +1) (proposed by Hosder et al. [35]) realizations of input stochastic vector Ξ = {ξ (1) , ξ (2) , · · · , ξ (N ) } is drawn. Next, for each realization the quantity of interest Y = {y (1) , y (2) , · · · , y (N ) }T , is evaluated where y (i) = U(ξ (i) ). Then, by solving the following overdetermined linear system with the least-squares approach, the coefficients are calculated.      ψ0 (ξ 1 ) · · · ψi (ξ 1 ) · · · ψP (ξ 1 ) u(ξ 1 ) u0  ..   ..   ..  .. .. .. ..  .  .   .  . . . .       ψ0 (ξ s ) · · · ψi (ξ s ) · · · ψP (ξ s )   ui  =  u(ξ s )  . (7)       ..   .   ..  .. .. .. ..  .   ..   .  . . . . N N N uP ψ0 (ξ ) · · · ψi (ξ ) · · · ψP (ξ ) u(ξ N ) For generating the sample points, the Sobol’ quasi-random sampling method is employed [36]. Since the PC basis are orthonormal, the mean and variance of the quantity of interest read as following [32], µ(x) = u0 (x), 11

(8)

σ 2 (x) =

P X

u2i (x).

(9)

i=1

5.2. Sensitivity Analysis Using Sobol’ Indices Quantifying the respective impacts of input stochastic variables (or combinations of it) onto the variance of a system response is defined as Sensitivity analysis (SA). The Sobol’ indices have received much attention due to the providing accurate information for most cases [37]. Each Sobol’ index Si1 ,...,is is a sensitivity measure that quantify which amount of the total variance is affected by the uncertainties in the set of input parameters {i1 , ..., is }. The PC-based Sobol’ indices are determined as:   X 1 Si1 ,··· ,is = 2  u2α hψα ψα i , (10) σ α∈Ii1 ,··· ,is

where Ii1 ,··· ,is is the set of α tuples such that only the indices (i1 , · · · , is ) are nonzero [37, 38]:   αk > 0 ∀k = 1, · · · , n, k ∈ (i1 , · · · , is ) Ii1 ,··· ,is = α : . (11) αk = 0 ∀k = 1, · · · , n, k ∈ / (i1 , · · · , is ) Furthermore, the total PC-based sensitivity indices are: X SjT1 ,··· ,js = Si1 ,··· ,is .

(12)

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5.3. Code Validation The UQ computation is validated in this section by performing UQ analysis on analytical Ishigami function (Ishigami and Homma [39]). The function is widely used as analytical test function for uncertainty and sensitivity analysis methods, because of its strong nonlinearity and non-monotonicity. The input uncertain parameters have uniform distributions. Hence uncertainties also can be quantified using askey-wiener PCE with Legendre polynomials. The Ishigami function is defined by: y = sin ξ1 + asin2 ξ2 + bξ34 sin ξ1

(13)

where the input variables ξi , i = 1, 2, ..., n are uniformly distributed over [−π, π]. The mean, variance and the Sobol sensitivity indices can be computed analytically: a µ= (14) 2 σ2 =

a2 bπ 4 b2 π 8 1 + + + 8 5 18 2 12

(15)

S1 =

1 σ2



bπ 4 b2 π 8 1 + + 5 50 2

 ,

S2 =

1 σ2



a2 8

 ,

S3 = 0

(16)

The numerical computation were conducted for {a = 7, b = 0.1} and the results are presented in Table 4. Comparing numerical and analytical results reveals that the UQ code successfully reproduced the statistics of the Ishigami function. Table 4: UQ results of Ishigami function

µ σ2 S1 S2 S3

Analytical

PCE

Relative error(%)

3.5 13.8446 0.3139 0.4424 0

3.5000 13.8450 0.3139 0.4424 0.0000

0.0000 0.0028 0.0000 0.0000 0.0000

6. Results and Discussion 250

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Assuming uncertainty in operational conditions given in Table 3, in this section the results of stochastic CFD computations using NIPC approach with polynomial order p = 3 are presented and discussed. First, the effects of presumed uncertainties on fluid flow characteristics including Mach number and pressure fileds are studied. Then, the stochastic thermal field under operational uncertainties is investigated. Finally, maximum temperature and temperature gradient of blade will be assessed. 6.1. Flow Field Around the C3X Turbine Vane

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Fig. 7 displays the contours of Mach number mean and Coefficient of Variation (CoV=σ/µ) of the flow around the blade on midspan plane. Both parameters take larger values downstream of the guide vane in comparison with upstream. As expected the mean Mach number is zero at the stagnation point which coincides with the leading edge of the vane. It is observed that flow accelerates along the suction side of the blade and the boundary layer transition occurs. Further downstream, a low Mach number zone is observed which is due to the upper vane wake. Close to the trailing edge of the vane, the Mach number is increased to about 1.05 indicating the presence of a transonic flow region with a weak shock. It is seen the velocity drops by approaching to the trailing edge wake. The high Mach number region close to the trailing edge of the pressure side of the vane is formed by the boundary layer transition of the bottom turbine vane. Concerning CoV variations shown in Fig. 7(b), it is seen that the Mach standard deviation of incoming flow is about three orders of magnitude smaller than the mean Mach. As consequence, it can be claimed that the uncertainties

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Figure 7: Contours of mean and Coefficient of variation of Mach number of the hot gas flow on the midspan plane.

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has no remarkable effects on the Mach number in the region upstream of the guide vane. On the other hand, at the downstream region, specially on the transonic area, higher standard deviations are observed. This is expected since both position and values of Mach number in transonic area are influenced by uncertainties in operational conditions. It is noted that these uncertainties affect the Mach field in the suction side boundary layer. The sensitivity analysis (SA) using Sobol’ indices is also performed in the present study to quantify the relative contributions of the input stochastic variables onto the variance of the outputs. Contours of total Sobol’ indices of Mach number at the midspan plane of vane are illustrated in Fig. 8. It is worth to mention that ST0,in , Slin , Sp0,in , SIin and Spout respectively stand for the Sobol’ indices of inlet total temperature, inlet turbulence length-scale, inlet total pressure, inlet turbulent intensity and outlet pressure. It is clearly visible that the Mach field variations around the blade are resulted from thermodynamic variables (primarily due to the uncertainties in inlet and outlet pressures and secondly caused by inlet temperature). A relatively uniform trend for the inlet total pressure Sobol’ index is observed around the blade. As expected, outlet pressure variations are more influential on the Mach field downstream of the blade. On the other hand, the Mach number in inlet region is affected by inlet total temperature uncertainties more than outlet region. Interestingly, turbulence quantities (the inlet turbulence length-scale and turbulence intensity) merely affect the Mach number close to solid walls and outside the boundary layers the Mach field can be considered independent from uncertainties in turbulence quantities. 14

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(e) Spout Figure 8: Contours of total Sobol’ indices of Mach number of the hot gas flow on the midspan plane.

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For detailed study of the flow field around the C3X vane, the stochastic pressure field is discussed in the following. In Fig. 9, the normalized two-dimensional PDF (PDF/max(PDF)) contour of pressure of the hot gas flow on blade outer surface is compared with the available experiments of the blade midspan plane. The results are presented at the blade midspan against normalized distance from the leading edge (normalized with the axial chord of the blade; xc =78.2 mm). A comparison is made on the mean, deterministic curves and experimental measurements. It can be seen that the mean and deterministic pressure distributions are similar and both predictions are in acceptable agreement with the measured values. The similarity between the mean and deterministic profiles can be explained by the fact that all random variables are assumed to have symmetric PDFs with mean values identical to the deterministic case. PDF

Deterministic

Mean(µ)

Experiment

1.1

1

1 0.8

p/p0,in

0.9 0.6

0.8 0.7

0.4

0.6 0.2 0.5 0.4

0

0.2

0.4

0.6

0.8

1

0

x/xc

Figure 9: Normalized fluid pressure on the blade outer surface at the blade midspan. Contours of normalized 2D PDF, mean and deterministic curves and experimental measurements are presented. 310

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Fig. 10 shows the contours of mean and CoV of static pressure coefficient around the blade on the midspan plane. Figs. 9 and 10(a) show a constant static pressure on the pressure side of the vane up to the midway between the leading and trailing edges of vane. Further downstream, a sudden decrease in pressure is observed which is caused by the flow acceleration in this zone. On the blade suction side, the pressure initially decreases up to about x/xc = 0.4 and thereafter, increases locally due to the effects of upper blade wake. Immediately after the local increase, the pressure is observed to descend gradually. Notice that only very small variations of the pressure around the blade are observed (Fig. 9). More specifically, in the majority of regions, the standard deviation of static pressure is about three orders of magnitude smaller than its mean (Fig. 10(b)) suggesting that the operational uncertainties can not significantly affect the pressure field. As anticipated, the effects of operational uncertainties on pressure variations at transonic region are more pronounced.

16

2 0.4

0.5

0.6

0.7

0.8

3

4

5

6

7

8 −3

0.9

x 10

(a) Mean

(b) Coefficient of variation

Figure 10: Contours of mean and Coefficient of variation of pressure coefficient of the hot gas flow on the midspan plane.

1

S o b o l’ I n d e x

0.8

ST 0,in Sl in Sp 0,in SI in Sp out

0.6

0.4

0.2

0 −1

0

−0.5

0.5

1

x/xc

Figure 11: Total Sobol’ indices of the fluid pressure on the blade outer surface at 50% of the blade span.

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In order to quantify the contribution of each random variable on pressure field, total Sobol’ indices on the blade surface are presented in Fig. 11. Negative x/xc values refer to the position on pressure side, while positive values correspond to positions on the suction side of vane. Fig. 11 clearly depicts that the randomness in inlet total pressure is the most influential parameter on the pressure distribution in −0.9 < x/xc < 0.4 which covers most of the pressure side as well as the suction side upstream on the transition point. For better understanding, contours of Sobol’ index of pressure field related to the inlet total pressure is shown in Fig. 12(a). As mentioned before, the flow acceler17

335

ates around the x/xc = 0.4 and transition from laminar to turbulent boundary layer occurs. Consequently, it is expected that the turbulence length-scale, turbulent intensity and total temperature play important roles on the location of the boundary layer transition. As expected, after transition region, similar to the region before x/xc = −0.9, variation of static pressure around the blade is mainly determined by the outlet pressure, which is shown with more details in Fig. 12(b).

0.2

0.4

0.6

0.8

0.2

(a) Sp0,in

0.4

0.6

0.8

(b) Spout

Figure 12: Contours of total Sobol’ indices of pressure of the hot gas flow on the midspan plane.

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6.2. Thermal Field For investigating the stochastic thermal field, the heat transfer coefficient at the blade surface, the temperature distribution on the blade and temperature field of hot gas around the vane are discussed respectively in this section. Fig. 13(a) shows contours of normalized two-dimensional PDFs of local heat transfer coefficient on the blade outer surface which is normalized with 2 the reference value reported in the literature [16] (1135W/m K) for the blade midspan plane. These contours are also compared with the mean and deterministic curves as well as the experimental data of [16]. Similar to the previous section, the mean and deterministic results of the local heat transfer coefficient at the blade surface are identical and a good consistency between both computations and experiments is found. Interestingly, the measurements lie within the non-deterministic PDF range, specifically in the region where PDF approaches to its peak values. This key finding clearly indicates that the discrepancies of numerical and experimental results can be partly due to the uncertainties in

18

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the experimental measurements. As reported above, a noticeable physical phenomena around the blade is the transition from laminar to turbulent boundary layer close to x/xc = 0.4. This feature significantly increases the heat transfer between the external hot fluid and the blade surface. It should be noticed that in transition region, the heat transfer coefficient increases abruptly and its PDF bandwidth is not clearly visible. Therefore, the CoV of heat transfer coefficient is shown in Fig. 13(b). As clearly depicted in this figure, the variation of heat transfer coefficient with respect to operational uncertainties increases considerably in transition region. At the trailing edge, the cooling channels are close to the blade outer surface, and thus fluctuations in local heat transfer coefficients in this region is visible. PDF

Mean(µ)

Deterministic

Experiment 1

0.11

1.2

Transition Region

0.1

0.8 1

0.09

0.6 CoVh

h/href

0.8 0.6

0.08

0.4 0.07

0.4 0.2 0.2 0−1

0.06

−0.5

0

0.5

1

0

x/xc

0.05 −1

−0.5

0

0.5

1

x/xc

(a) Normalized 2D PDF

(b) CoV

Figure 13: Normalized heat transfer coefficient of the blade outer surface at the blade midspan. (a) Contours of normalized 2D PDF, mean and deterministic curves and experimental measurements and (b) Coefficient of variation.

370

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The contours of mean and standard deviation of local heat transfer coefficient of the blade outer surface are presented in Fig. 14. This figure clearly demonstrates that the mean and standard deviation of local heat transfer coefficient downstream of the transition region is increased. Moreover, by moving from hub to tip, the coolant temperature rises which leads to lower heat transfer levels though the variation of heat transfer coefficient remains almost constant. Fig. 15 illustrates total Sobol’ indices of the blade outer surface heat transfer coefficient at the blade midspan plane. It is seen that the most influential operational variable in local heat transfer coefficient on the blade outer surface is the total inlet temperature. Furthermore, uncertainties in turbulence length-scale and turbulence intensity are dominant quantities influencing the variation of heat transfer coefficient in transition region. As discussed before, at x/xc = −0.4, flow field is influenced by the bottom vane transition zone. Consequently, though the effects of total inlet temperature on variations of heat transfer coefficient is reduced, turbulence quantities become remarkably influential. From Fig. 15 it can concluded that the local heat transfer coefficient is fairly robust with respect to the uncertainties in pressure conditions (i.e. total 19

(a) Mean

(b) Standard Deviation

Figure 14: Contours of mean and standard deviation of normalized heat transfer coefficient of the blade. 1

S o b o l’ I n d e x

0.8

ST 0,in Sl in Sp 0,in SI in Sp out

0.6

0.4

0.2

0 −1

0

−0.5

0.5

1

x/xc

Figure 15: Total Sobol’ indices of the blade outer surface heat transfer coefficient at the blade midspan.

385

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inlet pressure and outlet pressure). By moving from hub to tip the local heat transfer coefficient falls and is less affected by the total inlet temperature in suction side, specifically in the transition region. Nevertheless, in the pressure side which is affected by transition zone of bottom vane, the total inlet temperature becomes substantially influential. However, turbulence quantities follow totally reverse trends which is clearly illustrated in Fig. 16. The non-uniformity in spanwise direction specifically in the pressure side is correlated to the coolant temperature rise. The PDF contours of normalized temperature (Tref = 811K), together with mean, deterministic distributions and experimental data at the blade midspan are shown in Fig. 17. As it is observed, most of the measured data are within the band of temperature variation. Fig. 18 depicts the contours of mean and standard deviation of the blade surface temperature. Because of substantial increase of heat transfer coefficient at the transition region, noticeable decrease in

20

(a) ST0,in

(b) Slin

(c) Sp0,in

(d) SIin

(e) Spout Figure 16: Contours of total Sobol’ indices of heat transfer coefficient of the blade outer surface

400

the blade temperature is expected there. Since the cooling channels are located near the blade outer surface, oscillation in the temperature profile is observed close to the trailing edge. By increasing the coolant flow temperature from hub to tip, the blade surface mean temperature rises and leading and trailing edges temperatures experience 6% and 5% growth respectively. In addition, the variation of temperature is found to increase slightly indicating that the standard deviation increases along the coolant flow. Furthermore, due to the blade cross

21

PDF

Mean(µ)

Deterministic

Experiment

1

1

0.95 0.8

0.9

T /Tref

0.85

0.6

0.8 0.4

0.75 0.7

0.2

0.65 0.6 −1

−0.5

0

0.5

1

0

x/xc

Figure 17: Normalized temperature of the blade outer surface at the blade midspan. Contours of normalized 2D PDF, mean and deterministic curves and experimental measurements are presented.

(a) Mean

(b) Standard Deviation

Figure 18: Contours of mean and standard deviation of normalized temperature of the blade.

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section geometry, one expects to observe higher temperature level by approaching to the blade surface. Also, internal temperatures of blade at trailing edge and tip were found to be higher than those of leading edge and hub respectively. The same trend is also observed for standard deviation. To examine the effect of each uncertain parameter on blade surface temperature, total Sobol’ indices are presented in Figs. 19 and 20. These results clarify that the randomness in inlet total temperature has the main role on the variation of temperature as expected. Sensitivities indices related to the turbulence length-scale uncertainty increases at two regions namely, near the x/xc =-0.4 and 0.4, which leads to decrease in ST0,in . At x/xc =0.4 the turbulence quantities play an important role in this region due to boundary layer transition. The flow and thermal fields at x/xc =-0.4 on the pressure side are highly affected by the transition region of the bottom blade. Therefore, both Slin and SIin increase in this region. By increasing the coolant flow temperature from hub to tip, blade temperature increases and in the suction side, especially at 22

the transition region, the effect of inlet total temperature on blade temperature variations decreases while inlet turbulence length-scale and turbulent intensity have more pronounced effects. However, this trend is totally reversed in the pressure side. 1

0.8

S o b o l’ I n d e x

420

ST 0,in Sl in Sp 0,in SI in Sp out

0.6

0.4

0.2

0 −1

−0.5

0

0.5

1

x/xc

Figure 19: Total Sobol’ indices of the blade outer surface temperature at 10%, 50% and 90% of the blade span.

(a) ST0,in

(b) Slin

(c) SIin Figure 20: Contours of total Sobol’ indices of temperature of the blade

23

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430

To perform a deeper examination of the thermal field around the blade, contours of mean and CoV of temperature around the blade are depicted in Fig. 21. As anticipated, due to heat transfer of flow with blade, the hot gas temperature drops as it passes over the blade. The highest and the lowest temperature occur in stagnation point and transonic region respectively. The temperature around the vane is found vary mostly in downstream of the flow, specifically in transonic and transition region. This shows that the operational uncertainties have more impact on the temperature field in these zones.

0.8

0.85

0.9

0.95

0.0134 0.0136 0.0138 0.014 0.0142 0.0144 0.0146

(a) Mean

(b) Coefficient of variation

Figure 21: Contours of mean and Coefficient of variation of normalized temperature of the hot gas flow on the midspan plane.

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440

Contours of Sobol’ indices of temperature field is represented in Fig. 22. The most influential parameter is still total inlet temperature, although its effects are lightly reduced in transonic region. However, despite the blade surface temperature, the effects of total inlet pressure and outlet pressure on the variation of temperature field is higher than turbulent intensity and turbulence lengthscale, specifically in the transonic region. This can be attributed to the fact that transonic region is intrinsically influenced by pressure. It is worthwhile mentioning that these effects are adversely observed in the boundary layer and this might explain the differences of this trend and what happens to blade surface temperature.

24

1 0.96

0.97

0.98

2

3

2

3

4

5

6

7

5

8 −3

x 10

(a) ST0,in

1

4

0.99

(b) Slin

6

7

2

4

6

8

10

−3

12

14

16 −4

x 10

x 10

(c) Sp0,in

(d) SIin

25

0.005

0.01

0.015

0.02

(e) Spout Figure 22: Contours of total Sobol’ indices of temperature of the hot gas flow on the midspan plane.

PDF

Deterministic

PDF

Mean(µ)

Deterministic

Mean(µ)

0.8

0.02

0.7 0.6

0.015

0.5 0.4

0.01

0.3 0.2

0.005

0.1 0 700

710

720

730

740

750

760

770

0 12

780

12.5

13

13.5

14

14.5

|∇T |max (K/mm)

Tmax (K)

(b) |∇T |max

(a) Tmax

Figure 23: Probability distribution function of blade maximum temperature and temperature gradient, compared to mean and deterministic values.

6.3. Maximum Blade Temperature and Temperature Gradient

445

450

455

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In gas turbines, the high heat load is one of the most important factors of high-temperature components failure. It is commonly admitted that 20 K variation in the metals temperature, decreases the components life span by 50% (Han et al. [40]). Consequently, predictions have to consider this aspect. The effects of operational uncertainties on the maximum temperature and maximum temperature gradient in the investigated turbine vane, playing an important role in the determination of the blade lifespan, are presented and discussed in this section. Probability distribution function of blade maximum temperature is depicted in Fig. 23(a). The mean and deterministic values of Tmax are also compared in this figure. In a wide range of the band of predicted PDFs of maximum blade temperature, it appears to be almost constant. As illustrated, there is no difference between mean and deterministic values of Tmax . Fig. 23(b) represents the PDF curves, mean and deterministic values of the maximum temperature gradient of the blade. It is noticed that the |∇T |max PDF is relatively uniform too. Moreover, the values of mean and deterministic are approximately the same. In order to investigate the robustness of the blade lifespan under stochastic condition, the non-deterministic data of Tmax and |∇T |max , normalized with their corresponding mean values, are represented in Fig. 24 in the form of boxplots. On each box, the lower and upper edge of the box demonstrates the first (25%) and third (75%) quartiles, while the median represented by central red line. Dashed lines indicate the range of the data variation outside the first and third quartiles. The 1.5 interquartile range of the lower and upper quartiles is showed by the ends of dashed lines. This figure suggests considerable interquartile ranges for both Tmax and |∇T |max . The sensitivity indices of maximum temperature and maximum temperature gradient are exhibited in the following figures. Fig. 25(a) demonstrates the first and total Sobol’ indices of Tmax in the form of bar chart. It is obvious that 26

1.04 1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 0.95

|∇T |max

Tmax

Figure 24: Box plots of blade maximum temperature and maximum temperature gradient normalized with their mean values

470

the effect of uncertainties in inlet total pressure, outlet pressure, and turbulence intensity are negligible and the maximum temperature is mostly affected by inlet total temperature. The first and total Sobol’ indices are similar indicating insignificance of interaction terms in the PCE of Tmax . Fig. 25(b) displays the same results for maximum temperature gradient of the turbine vane. 1

First Total

0.8

Sobol’ indices of |∇T |max

Sobol’ indices of Tmax

1

0.6 0.4 0.2 0

S1

S2

S3

S4

0.6 0.4 0.2 0

S5

First Total

0.8

S1

S2

S3

S4

S5

(b) |∇T |max

(a) Tmax

Figure 25: First and total Sobol’ indices of maximum temperature and temperature gradient of the balde.

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7. Conclusion In this paper, uncertainty quantification of full conjugate heat transfer simulation of cooled NASA C3X turbine blade was presented and discussed. The uncertainties of flow conditions including the inlet total pressure, total temperature, turbulent intensity, turbulence length-scale and outlet static pressure are set equal to experimental measurements errors. It was found that the uncertainties in operating conditions could explain the discrepancy between deterministic and experimental results. The stochastic condition causes significant variations

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in the flow and thermal fields around the blade as well as the blade temperature profile especially in transition as well as transonic region and flow boundary layer. Sensitivity analysis indicates that the uncertainty in inlet total pressure and outlet static pressure have the utmost contributions for variations in pressure and Mach fields. In contrast, the temperature and heat transfer coefficient on the outer surface of the blade are mainly affected by the inlet total temperature and the turbulence length-scale. Obviously, the main stochastic parameter that causes significant variation for all measured quantities at the transition region is turbulence length-scale. Thus, it is necessary to assume the proper value and its uncertainties for this parameter. Moreover, the stochastic conditions can lead to an important variation on the turbine vane temperature profile including maximum temperature and temperature gradients of the blade that changes up to 30 K and 7.5 K/cm respectively. These changes could cause the noticeable decrease in blade lifespan. References [1] R. Ghanem, P. Spanos, Stochastic finite elementsa spectral approach, 3rd Edition, Springer, New York, 1938. doi:10.1007/978-1-4612-3094-6.

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Highlights:     

Stochastic conjugate heat transfer simulation of a gas turbine vane is performed Operational uncertainties in thermodynamic and turbulence conditions are considered Uncertainties cause significant variations in the hydrodynamic and thermal fields Most of the experimental data are covered with the band of each parameter variation Most effective stochastic parameters are found using sensitivity analysis