Evaluation of Different Turbulence Models and Numerical Solvers for a ...

J. Energy Power Sources Vol. 1, Number 1, 2014, pp. 17-30 Received: July 2, 2014, Published: July 25, 2014

Journal of Energy and Power Sources www.ethanpublishing.com

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade Amgad M. Abbass1, Medhat M. Sorour2 and Mohamed A. Teamah2 1. Maintenance Planning Sector, Abu Qir Fertilizers Co., El-Tabia Rashid road, Alexandria, Egypt 2. Mechanical Eng. Dept., Alexandria University, Alexandria, Egypt Corresponding author: Amgad M. Abbas ([email protected]) Abstract: The gas path over the turbine blades is a very complex flow field due to the variation of flow regime and the corresponding heat transfer, This investigation is devoted to study the two and three-dimensional predictive modeling capability for airfoil external heat transfer by using pressure based solver PBS and density based solver DBS. The results show the effects of strong secondary vortexes, laminar-to-turbulent transition, and also show stagnation region characteristics. Simulations were performed on an irregular quadratic two and three-dimensional grids with the Fluent 6.3 software package employing several turbulence models (Spalart-Allmaras, RNG k-Є and SST k-ω model). Detailed heat transfer predictions are given for a power generation turbine rotor with 127° of nominal turning, axial chord of 130 mm and blade aspect ratio of 1.17. The comparison was made with the experimental and the numerical results of Giel et al. and a good agreement was found with the DBS and Spalart-Allmaras turbulent model and it has been concluded in particular that rather fine computational three dimensional grids are needed to get accurate local heat transfer controlled by complex 3D structure of secondary flows. Key words: Transonic gas turbine, turbulent condition, pressure based solver PBS, density based solver DBS.

Nomenclature:

Greek Symbols

Cx d h K k M Nu P Pr Q

Blade axial chord (m) Leading edge diameter (m) Heat transfer coefficient (W/m2-K) Thermal conductivity (W/m-K) Turbulent kinetic energy Mach number (dimensionless) Nusselt number (dimensionless) Pressure (Pa) Prandtl number (dimensionless) Heat flux (W/m2)

Reex

Reynolds number, Reex =

r S T Tu U Y+ Z

1 3

ρUex Cx μ

(dimensionless)

Recovery factor, r = Pr Blade surface length (m) Temperature (K) Turbulent Intensity (%) Free-stream velocity (m/s) Dimensionless wall distance Spanwise coordinate, normalized by blade span

γ Є Λx μ ρ ω Subscripts o in ex Isen av aw

Ratio of specific heats, (dimensionless) Turbulent Dissipation Rate (m2/s3) Length scale (mm) Dynamic viscosity (kg/s.m) Local Density (kg/m3) Specific dissipation rate (s-1) Total condition Inlet free stream value Exit free steam value Isentropic value Average value Adiabatic wall value

1. Introduction It is well known from the thermodynamic analysis that the performance of a gas turbine engine is strongly influenced by the temperature at the inlet to the turbine.

18

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

There is thus a growing tendency to use higher turbine inlet temperatures, implying increasing heat loads to the engine components. Modern gas turbine engines are designed to operate at inlet temperatures of 1800-2000 K, which are far beyond the allowable metal temperatures. Thus, to maintain acceptable life and safety standards, the structural elements need to be protected against the severe thermal environment. So that as turbine inlet temperature increase, the necessity for accurate heat transfer predictions also increases, and for accurate heat transfer predictions.

naphthalene sublimation method. They showed that significantly different patterns are observed on the blade surface, especially near the blade tip due to the variation in tip leakage flow, and the heat/mass transfer characteristics on the blade surface are affected strongly by the local flow characteristics, such as laminarization after flow acceleration, flow transition, separation bubble and tip leakage flow. Ref. [3] studied the Effect of rotation on detailed film cooling effectiveness distributions in the leading edge region of a gas turbine blade, with three

With efficiency and power increases of modern gas

showerhead rows of radial-angle holes were the

turbines, researchers tried continuously to increase the

effectiveness measured using the Pressure Sensitive

inlet temperature to the maximum. This can be done

Paint (PSP) technique. Tests were conducted on the

only with better blade cooling, great heat transfer

first-stage rotor blade. The effect of the blowing ratio

comprehension and three-dimensional distribution of

was also studied. They showed that the different

the temperature inside the turbine. To give a detailed

rotation speeds significantly change the film cooling

cooling analysis as well as a good thermal structure of

traces with the average film cooling effectiveness in the

blades, several researchers treated experimentally and

leading edge region increasing with blowing ratio.

numerically in this area as in the following Experimental

blades using air or steam as coolants. Three main

experimentally the airfoil external heat transfer to

schemes of the blade cooling were used, namely

engine specific conditions including blade shape,

air-cooling, Open-Circuit Steam Cooling (OCSC) and

Reynolds numbers, and Mach numbers. Data were

Closed-Loop Steam Cooling (CLSC). The results

obtained in steady-state using a thin-foil heater

showed that steam appears to be a potential cooling

wrapped around a low thermal conductivity blade with

medium, when employed in an open-circuit or in a

127 deg of nominal turning and an axial chord of 130

closed-loop scheme. The combined system with CLSC

mm. Surface temperatures were measured using

gives better overall performance than does air-cooling

calibrated liquid crystals. The results showed the

or the OCSC.

of

strong

Ref.

secondary

[1]

Ref. [4] made a heat transfer analysis of gas turbine studied

effects

studies:

vortical

flows,

Ref. [5] studied the effect of hole angle and hole

laminar-to-turbulent transition, and also show good

shaping in the turbine blade showerhead film cooling.

detail in the stagnation region.

The film cooling measurements were presented on a

Ref. [2] studied experimentally the local heat/mass

turbine blade leading edge model with three rows of

transfer characteristics on the stationary blade near-tip

showerhead holes. They showed that, the shaping of

surface for various relative positions of the blade. A

showerhead holes provides higher film effectiveness,

low speed wind tunnel with a stationary annular turbine

than just adding an additional compound angle in the

cascade was used. The test section has a single turbine

transverse

stage composed of sixteen guide plates and sixteen

effectiveness than the baseline typical leading edge

blades. Detailed mass transfer measurements were

geometry. Numerical studies: Ref. [6] studied the Heat transfer on a film-cooled rotating blade using a multi-block,

conducted for the stationary blade fixed at six different relative blade positions in a single pitch using a

direction

and

significantly

higher

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

three-dimensional Navier-Stokes code to compute heat transfer coefficient on the blade, hub and shroud for a rotating high-pressure turbine blade with 172 film-cooling holes in eight rows. Wilcox’s k-ω model was used for modeling the turbulence. Of the eight rows of holes, three are staggered on the shower-head with compound-angled holes. They showed that the heat transfer coefficient is much higher on the blade tip and shroud as compared to that on the hub for both the cooled and the uncooled cases. Ref. [7] performed a computation to predict the three-dimensional flow and heat transfer of concave plate that is cooled by two staggered rows of film-cooling jets. This investigation considers two coolant flow orientations. They showed that the laterally averaged film-cooling effectiveness over the concave surface with a straight-blow plenum is slightly higher than that of a cross-blow plenum at all test blowing ratios and the blowing ratio is one of the most significant film-cooling parameters over a concave surface. Ref. [8] used the v2-f models of Durbin and the new v2-f model of Jones et al. to compute the blade heat transfer for the transonic cascade of Giel et al. Three different experimental cases were considered. They showed that the v2-f model of Durbin provides good predictions on the pressure surface. But there is some discrepancy between the suction surface heat transfer predictions and experimental measurements. The newer model of Jones et al. requires further

the suction side of the blade. Use of the SST model does require the computation of distance from a wall, which for a multi-block grid, such as in This case, can be complicated. Ref. [10] studied numerically the transonic gas turbine blade-to-blade compressible fluid flow, and determined the pressure distribution around the blade and compared the characteristic flow effects of Reflecting Boundary Conditions (RBC) and Non-Reflecting Boundary Conditions (NRBC) newly implemented in FLUENT 6.0. They showed that the modified k-Є, the Spalart-Allmaras and the RSM models gives good results compared with experiment, instead to use all these models to determine the effects of reflecting and non-reflecting boundary conditions only the RNG model is selected, also this model with NRBCs formulation is used to see the effects of the inlet turbulence intensities the exit Mach numbers and the Inlet Reynolds numbers. The results are closest to experiment. The blade geometry used in the present study is representative of first stage turbine blade for a new GE heavy frame power turbine machine design and presented in Fig.1.The turbine blade is for a machine operating in the 1370℃ (2500℉) class. The full power design point isentropic pressure ratio of the current blade section is 1.443. The inlet Mach number is 0.399 and the exit isentropic Mach number is 0.743. The inlet angle of attack is 59.1 deg while the exit angle is 67.9 deg, producing an aggressive total turning of 127 deg.

modification before it can be routinely used for blade heat transfer. Ref. [9] used two versions of the two-equation k-ω model and a Shear Stress Transport (SST) model with a three-dimensional, multi-block, Navier-Stokes code to compare the detailed heat transfer measurements on a transonic turbine blade. They showed that the SST model resolves the passage vortex better on the suction side of the blade, thus yielding a better comparison with the experimental data than either of the k-ω models. However, the comparison is still deficient on

19

Fig. 1

Blade geometry.

20

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

The blades linear cascade was used in this study

well-known is the k-ε model [12]. In turbulence models

because of the previous studies have shown that rotor

that employ the Boussinesq approach, the central issue

geometries in linear cascades provide good mid span

is how the eddy viscosity is computed. The model [13]

data as compared to their rotating equivalents.

solves a transport equation for a quantity that is a

The objective of the present study is to present the results of predicted heat transfer of a transonic turbine blade cascade obtained using ANSYS-FLUENT 6.3 CODE. One of the goals is to compare between the solution which obtained using two different numerical solvers such as Pressure Based Slover (PBS) and density based solver (DBS) and study capability of a three different turbulent models Spalart-Allmaras model, RNG k-Є model and SST k-ω model.

modified form of the turbulent kinematic viscosity. 2.1 Spalart-Allmaras Model The transported variable in the Spalart-Allmaras model, v is identical to the turbulent kinematic viscosity except in the near-wall (viscous-affected) region. The transport equation for v: ∂ ∂ ρv + ρ vui = Cb1 ρS v + ∂xi ∂t

2. Governing Equations Reynolds-Averaged Navier-Stokes equations are the governing equations for the problem analyzed (momentum balance), with the continuity equation. For the two and three-dimensional compressible flow of a Newtonian fluid, mass and momentum equation

1

∂

σv ∂xj

∂

ρui = 0

(1)

∂xj

∂xj

In Eq. (2),

+

∂uj 2

∂ul

3

∂xl

∂xi

-Cw1 ρfw

∂xj

- δij

-ρ u'i u'j

+

∂ ∂xj

-ρ u'i u'j

(2)

is the additional term of

X ≡

v

(7)

v

fv2 = 1-

X

(8)

1+X fv1

1+C6w3

1

6

has to be modeled for the closure of Eq. (2). The classical approach is the use of Boussinesq hypothesis,

g = r+Cw2 (r6 -r)

formulation as proposed by Kolmogorov, Boussinesq hypothesis is written as: -ρ u'i u'j = μt

∂ui ∂xj

+

∂uj ∂xi

-

2 3

ρk+μt

∂μk ∂xk

δij

(3)

(4)

6)

X + C3v1 3

fw = g

through the eddy viscosity concept [11]. In its general

d

(5)

X3

Reynolds stresses due to velocity fluctuations, which

relating Reynolds stresses and mean flow strain,

v 2

Where fv1 =

∂ ∂ ∂p ρui + ρui uj = + ∂t ∂xj ∂xi ∂ui

∂xj

2

∂v

+Cb2 ρ

μt = ρvfv1 ∂xi

μ

∂v

Note that since the turbulence kinetic energy k is not calculated in the Spalart-Allmaras model, the last term in Eq. (3) is ignored when estimating the Reynolds stresses. The eddy viscosity is given by

become, respectively

∂

μ+ρ v

g6 +C6w3

≡ S ≡ S+

v Sκ2 d2 v f κ2 d2 v2

(9) (10) (11) (12)

Where S is the magnitude of the vorticity and d is the distance from the nearest wall. Acknowledged that one

Successful turbulence models are those based on the

should also take into account the effect of mean strain

eddy viscosity concept, which solve one and two and

on the turbulence production, and a modification to the

seven scalar transport differential equations. The most

model has been proposed in Ref. [14] and incorporated

21

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

into the present study. This modification combines

μt = ρCμ

measures of both rotation and strain tensors in the definition of S: S ≡ Ωij + Cprod min 0, Sij - Ωij

(13)

Ωij ≡ 2Ωij Ωij , Sij ≡ 2Sij Sij

(14)

The mean strain rate, Sij defined as: 1 ∂uj ∂ui

Sij =

2 ∂xi

-

(15)

∂xj

The mean rate-of-rotation tensor, Ω defined as Ωij =

1 ∂ui ∂uj 2 ∂xj

-

(16)

∂xi

And the constants are Cb1 = 0.1355 , Cb2 = 0.622 , σv = 2/3 , Cv1 = 7.1 , Cw1 = Cw3 = 2 , κ

=

Cb1 k2

0.4187 and

+

1+Cb2 σv

, Cw2 = 0.3 ,

Cprod = 2.0

which

μt = μt0 f αs , Ω ,

∂ ∂ ρk + ρkui = ∂t ∂xi

calculated without the swirl modification using Eq. (19). Ω is a characteristic swirl number evaluated within FLUENT, and αs is swirl constant and set to 0.07 for mildly swirling flow. This swirl modification always takes effect for axisymmetric, swirling flows and three-dimensional flows when the RNG model is selected. And C*2ϵ is given by:

αk μeff

∂k ∂xj

+Gk +Gb -ρϵ-YM

(17)



∂xj

αϵ μeff

∂xj

Cμ η3 1-η/η0

(21)

1+βη3

+C1ϵ

k

Gk +C3ϵ Gb -C*2ϵ ρ

The eddy viscosity is given by

k

(22)

ϵ

And the production of turbulence kinetic energy Gk is given by: Gk = μt S2 (23) Where S is the modulus of the mean rate-of-strain tensor, defined as: S = 2Sij Sij (24) Where Sij Calculated from Eq. (15) The generation of k due to buoyancy in Eq. (17), and the corresponding contribution to the production of Є in Eq. (18). Where Gb for ideal gases is given by: μt ∂ρ

(25)

ρPrt ∂xi

For the standard and realizable k-Є models, the value of Prt is constant value of 0.85. In the case of the 1

RNG k-Є model, Prt = , where α is given by:

∂ ∂ ρϵ + ρϵui = ∂xi ∂t ϵ

η = S

Gb = -gi

And

∂ϵ

(20)

ϵ

Where

The RNG-based k-Є model is derived from the instantaneous Navier-Stokes equations, using a mathematical technique called “Renormalization Group” (RNG) methods. The analytical derivation results in a model with constants different from those in the standard k-Є model, and additional terms and functions in the transport equations for k and Є. A more comprehensive description of RNG theory and its application to turbulence can be found in Ref. [15]. The transport equations for the RNG k-Є model have a similar form to the standard k-Є model:

∂

K

Where μt0 is the value of turbulent viscosity

C*2ϵ = C2ϵ +

2.2 RNG k-Є Model

∂xj

(19)

ϵ

And after adding the swirl modification the eddy viscosity becomes

represented in Ref. [14].

∂

k2

α

α-1.3929 ϵ2 k

0.6321

α0 -1.3929

(18)

α+2.3929 0.3679 α0 +2.3929

=

μmol μeff

(26)

Where α0 =

1 Pr

=

k μCp

(27)

22

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

The Є is affected by the buoyancy is determined by the constant C3Є. In FLUENT, C3Є is not specified, but is instead calculated according to the following relation which proposed in Ref. [16]: C3ϵ = tanh

v

(28)

u

Where v is the component of the flow velocity parallel to the gravitational vector and u is the component of the flow velocity perpendicular to the gravitational vector. And the effect of compressibility according to a proposal Ref. [17] (YM) is given by: YM = 2ρϵM2t

Γk = μ+ Γω = μ+

σk = σω =

k

Φ1 = min max

(31)

(

μmol μeff

≪ 1

),

∝k = ∝ϵ ≈ 1.393. The model constants are C1ϵ = 1.42 , C2ϵ = 1.68 , Cμ = 0.0845, η0 = 4.38, β = 0.012.

The SST k-ω model [18] has a similar form to the standard k-ω model [19] where the turbulence kinetic energy, k, and the specific dissipation rate, ω are obtained from the following transport equations: ∂t

ρk +

∂

ρkui =

∂xi

∂ ∂xj

Γk

∂k ∂xj

+Gk -Yk +Sk

μt =

(38)

1 1 ∂k ∂ω σω,2 ω ∂xj ∂xj

,

4ρk σω,2 D+ω y2

, 10-10

(39) (40)

ρk

1

ω max

(41)

1 SF2 , α* a1 ω

Where the coefficient ∗ damps the turbulent viscosity causing a low-Reynolds-number correction and given by: α* = α*∞

α*0 + Ret /Rk 1 + Ret /Rk

Ret = α*0 = And

(42)

Γω

∂ω ∂xj

+Gω -Yω +Dω +Sω

ρk

(43)

μω

βi

(44)

3

βi = 0.072

(45)

F2 = tanh Φ22

(46)

is given by:

(32)

Φ2 = max 2

∂ ∂ ρω + ρωui = ∂t ∂xi ∂xj

(37)

by:

And

∂

1 F1 1-F + 1 σω,1 σω,2

Where

2.3 SST k-ω Model

∂

(36)

500 μ √k , 0.09 ω y ρy2 ω

D+ω = max 2ρ

And the inverse effective Prandtl numbers, αk andαϵ , are computed using Eq. (27) where α0 = 1 and in the limit

1 F1 1-F + 1 σk,1 σk,2

Where μt is the turbulent eddy viscosity and given

Where a is the speed of sound and given by:

high-Reynolds-number

(35)

F1 = tanh Φ41

(30)

a = γRT

μt σω

Where F1 and F2 are given by:

(29)

a2

(34)

Where Sis the strain rate magnitude and the turbulent Prandtl numbers for k and ω is given by

Where Mt is the turbulent Mach number and defined by: Mt =

μt σk

√k 0.09 ω y

,

500 μ ρy2 ω

(47)

Where, y is the distance to the next surface. (33)

Where the effective diffusivities for the k-ω model are given by:

The production of k (

) is given by:

Gk = min Gk , 10ρβ* kω

(48)

Gk = μt S2

(49)

Where

23

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

β* = β*i 1+ ζ* F(Mt )

β*i = β*∞

4 + 15

1+

4

Ret Rβ Ret Rβ

(50)

(51)

4

And the compressibility function F(Mt) is given by: 0 Mt ≤Mt0 (52) F(Mt) = 2 2 Mt -Mt0 Mt >Mt0 Where 2k

M2t =

(53)

a2

Where (a) is the sonic speed calculated from Eq. (31) and the production of ω (Gω ) is given by: ω

Gω = α Gk

(54)

k

∝∞ ∝0 +Ret /RW ∝*

(55)

1+Ret /RW

Where ∝∞ is given by: α∞ = F1 α∞,1 + 1-F1 α∞,2

(56)

Where α∞,1 =

βi,1 β*∞

-

κ2 σ ω,1

, α∞,2 = β*∞

βi,2 β*∞

-

κ2 σ ω,2

(57) β*∞

Where τij

eff

Keff

∂T ∂xj

+ui τij

Yk = ρβ* kω

is the deviatoric stress tensor and

given by: τij

eff

= μeff

∂uj ∂xi

+

∂ui ∂xj

2

- μeff 3

∂uk

δ ∂xk ij

(63)

For Spalart-Allmaras and SST-kω models, the effective thermal conductivity is given by: Cp μt Prt

(64)

Where, a constant value of 0.85 is used for the turbulent Prandtl number. But for the RNG, the effective thermal conductivity is given by: Keff = ∝Cp μeff

(65)

4. Numerical Methods inlet zone located at 15.24 cm from the leading edge; it

(58)

The Dissipation of ω (Yω ) is given by: 2

(62)

eff

The study domain consists of one passage with an

The Dissipation of k (Yk ) is given by:

Yω = ρβω

∂ ∂xj

Keff = K+

Where ∝ =

In FLUENT, turbulent heat transport is modeled using the concept of Reynolds’ analogy to turbulent momentum transfer. The “modeled” energy equation is thus given by the following: ∂ ∂ ρCP T + ρui CP T+1 = ∂t ∂xi

is limited by the blades pitch length 13 cm. The blade profiles were obtained by creating real edges from a given 143 points defining the blade geometry.

(59)

GAMBIT forms the edges in the shape of general NURBS curve of degree n which is GAMBIT employs

Where β* β = βi 1- βi ζ* F i

a value of n = 3 Fig. 1 (NURBS is a specific notation to (60)

Gambit it is describes how to create real curve from

Instead of a having a constant value, βi is given by:

existing points). The irregular quadratic semi structured grid is generated by the pre-processor GAMBIT (v2.3.16). According to this complex geometry, the grid is obtained using the QUAD PAVE scheme for the 2-d mesh which used in the 2D case as shown in Fig. 2 and with total number of cells as shown in Table 1 , the grid which used in the 3d case is obtained using the 2D grid as the base of the 3D grid volume , the 3D

Mt

βi = F1 βi,1 + 1-F1 βi,2

(61)

And the constants are σk,1 = 1.176, σω,1 = 2, σk,2 = 1, σω,2 = 1.168 , a1 = 0.31 , βi,1 = 0.075 , βi,2 = 0.0828 , α*∞ = 1, α0 = 1/9, β*∞ = 0.09, Rβ = 8, Rk = 6, Rω = 2.95, κ = 0.41, ζ* = 1.5, Mt0 = 0.25.

3. Convective Heat Transfer

24

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

grid was generated using the HEX-WEDGE COOPER scheme with 30 cells in span wise direction as shown in Fig. 3, the total number of cells are as shown in Table 1. Near-wall grid is ‘‘adapted” using the boundary layer method as to capture the important characteristics of the turbulent flow and to make the grid independence. Adaptation refines the grid in both streamwise and spanwise directions. The refined grid along the wall region reduces the first y+ to lower than value of 5. The grid independent study was conducted to achieve a very small change in average Nusselt number on turbine blade suction and pressure side. The commercial software package FLUENT (V. 6.3) from ANSYS, Inc. is adopted in this study. Fluent employs a finite-volume method with second order upwind scheme for spatial discretization of the convective

terms. The initial values were taken from the inlet boundary condition. We made use of “Coupled solution method” [20]. Using this approach, the governing equations are solved simultaneously, i.e., coupled together. Governing equations for additional scalars will be solved sequentially. A control volume-based technique is used that consists of: • Division of the domain into discrete control volumes using a computational grid; • Integration of the governing equations on the individual control volumes to construct algebraic equations for the discrete dependent variables (unknowns) such as velocities, pressure, temperature, and conserved scalars; • Linearization of the discretized equations and solution of the resultant linear equation system to yield updated values of the dependent variables. The coupled approach is designed for high speed compressible flows and gives very satisfactory results in turbo machines especially with the implicit coupled solver. The second order upstream [20] scheme is used because higher-order accuracy is desired.

5. Boundary Conditions The boundary conditions are very important to Fig. 2

The computational grid of two dimensional models.

Table 1

Grid size parameters values. Cells Faces 2D Mesh 7687 15526 3D Mesh 215236 657651

Nodes 7934 230086

obtain an exact solution with a rapid convergence. According to the theory of characteristics, flow angle, total pressure, total temperature, and isentropic relations are used at the subsonic inlet. The same relations are used to determine the fluid properties at the supersonic outlet such as static pressure, static temperature and the isentropic exit Mach number. All of the fluid properties which used as a boundary condition for the present study cases are described below and in Table 2: • All the walls (pressure and suction sides) are considered Isothermal 345oK with no slip condition, the temperature of the unheated endwall was set equal

Fig. 3 models.

The computational grid of three dimensional

to the inlet total temperature and mid span symmetry was assumed;

25

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade Table 2 Reex (%)

Boundary conditions of the present cases. Reex

Min

Po in

Pin

Pex

100

2488000

0.352

156790.9

143896.4

108776.8

75

1866000

0.352

117593.2

107922.3

81582.57

50

1244000

0.352

78395.4

71948.19

54388.38

30

746400

0.352

47037.2

43168.92

32633.03

• Since the periodic boundaries are satisfied, instead of modeling the whole blades row of 11 passages of Ref. [1], only one passage is used in order to limit the computational time and costs; • Total pressure, total temperature 300oK, inlet total and static pressure, inlet flow angle and inlet Tu = 9% and ΛX = 0.026 m, are specified as the inlet conditions; • Outlet static pressure according to Mex = 0.742 and outlet flow angle is specified as the outlet conditions.

6. Nusselt Number

measured and another numerical data. These distribution were obtained with 100% Reex Case and Pr = 0.7. generally can concluded that all models capture the main trends and no significant difference between the turbulence models and the numerical solvers which used but the 3D models gives more accurate results near the suction side trailing edge as shown in Figs 6-7. Figs. 8-9 show isentropic Mach number contours over the blade span for the 3D model computed with three different turbulence models using the PBS and DBS in comparison with another numerical data. These

The Nusselt number was defined as follows: Nu =

h.Cx

1.0

Where is the heat transfer coefficient h is defined as:

0.8

h =

Qw

(67)

(Tw -Taw )

Misen

(66)

K(To in )

0.4

Where the local adiabatic wall temperature Taw is defined as: Taw = r +

1 - r 1 + 0.5 γ - 1 M2isen

*To in

r = Pr

2d -P B S . R N G k- ε 2d -P B S . S st k- ω 2d -PBS . Spalart-A. G iel [11] :C alc. G iel [11] :E xpr.

0.2

(68)

0.0 -1.0

In expression (3) the local isentropic Mach number Misen, is determined from the wall static pressure, and the recovery factor r is evaluated as: 1 3

0.6

-0.5

0.0

S/C x

0.5

1.0

1.5

2.0

Fig. 4 Comparison of Misen over the blade mid span using three different turbulence 2D models and PBS.

1.0

(69) 0.8

K(To in )

(70)

0.6

Misen

Pr =

μ Cp

7. Results and Discussion

0.4

Figs. 4-7 show isentropic Mach number distribution over the blade mid span for the 2D and 3D models computed with three turbulence models and with two different numerical solvers (Pressure based solver PBS and Density based solver DBS ) in comparison with the

0.2

G ie l [1 1 ] G ie l [1 1 ] 2d- D BS 2d- D BS 2d- D BS

: C a lc : Exp. S STk−ω R N G k −ε S p a la rt-A .

0.0 -1 .0

-0 .5

0.0

0 .5

1 .0

1 .5

2 .0

S /C x

Fig. 5 Comparison of Misen over the blade mid span using three different turbulence 2D models and DBS.

26

Evalluation of Diffferent Turbule ence Models and a Numerica al Solvers for a Transonic Turbine Blade Cascade

1.0

0.8

Misen

0.6

0.4

3d - PBS. RNG k-ε 3d - PBS. Sst k-ω 3d - PBS. Spalart-A. Gie el [11] :Calc. Gie el [11] :Expr.

0.2

(cc) 3D-PBS Misenn contours usingg RNG K-Є moodel

0.0 -1.0

-0.5

0.0

0.5

1.0 0

1.5

2.0

S/C x

Fig. 6 Compaarison of Misen over the bladee mid span usin ng three differentt turbulence 3D D models and PB BS. 1.2

1.0

Misen

0.8

0.6

(d d) 3D-PBS Miseen contours usin ng SST k-ω mod del 0.4

Fig. 8 Misen contours over the blade b span using three nt turbulence 3D models and PBS. P differen

Giel [11] : Calc G G [11] : Exp. Giel 3d d-DBS. SSTk−ω 3d d-DBS. RNGk−ε 3d d-DBS. Spalart-A.

0.2

0.0 -1.0

-0.5

0.0

0.5

1.0

1.5

2.0 0

S/Cx

Fig. 7 Compaarison of Misen over the bladee mid span usin ng three differentt turbulence 3D D models and DBS.

(a) Ref. [1] Misen contours off numerical datta over the blad de i span

(b) 3D-PBS Misen contours u using Spalart-A Allmaras modell

contouurs obtained with w 100 % Reeex Case and P Pr = 0.7. All turrbulence mod dels and solvvers gives verry good agreem ment with the another num merical resultss except the RN NG k-Є mo odel with thhe PBS givees some exceptiions in data ov ver the suction side. From m all figures which w presents the isentropic Mach numbeer concluded that no decellerating flow regions are seeen on the suctiion surface unntil near the geeometric throat at S/Cx = 1.07 where verry slight deceeleration occur on the uncovered porrtion of the blade. Deceleeration also seen s on the pressure p surfface just downsttream of the leading l edge, extending too S/Cx = -0.35. The calculateed values Missen were usedd for the heat traansfer parameeters calculatioon. Figss. 10-13 show local Nu num mber distributiion over the blade mid span for fo the 2D and 3D models coomputed with thhree turbulencce models annd with two ddifferent numeriical solvers PBS and DBS ) in comparisson with the measured m and another num merical data. These distribuution were ob btained with 1000% Reex casee and Pr = 0.7.

27

Evalluation of Diffferent Turbule ence Models and a Numerica al Solvers for a Transonic Turbine Blade Cascade 6500

G iel [11] G iel [11] 2d-D B S 2d-D B S 2d-D B S

6000 5500 5000

: E xp. : C alc S p alart-A . R N G k −ε S S Tk −ω

4500

Nu

4000 3500 3000 2500 2000

(a) 3D-DBS Misen contours for Spalart-Alllmaras model

1500 1000 -1.0

-0.5

0.0

S/C x

0.5

1 .0

1.5 5

2.0

Fig. 11 Comparison n of the local Nu N over the bllade mid span ussing three differrent turbulencee 2D models an nd DBS. Giel [11] Giel [11] 3d-PBS. 3d-PBS. 3d-PBS.

6000

DBS Misen contoours for RNG k-Є k model (b) 3D-D

: Exp. : Calc. Spalart A. S R RNG k-ε S SST k-ω

Nu

4000

2000

-1.0

-0.5

0.0

0.5

1.0

1.5 5

2.0

S/Cx

Fig. 12 Comparison n of the local Nu N over the bllade mid nd PBS. span ussing three differrent turbulencee 3D models an

(c) 3D-D DBS Misen contours for SST k--ω model Fig. 9 Misen contours overr the blade sp pan using threee ulence 3D modeels and DBS. different turbu

7000

G ie l [1 1 ] G ie l [1 1 ] 3 d -D B S . 3 d -D B S . 3 d -D B S .

6500 6000 5500 5000

Gi el [11] : Exp. Gi el [11] : C alc. 2d d PBS. Spalart A. 2d d PBS. R NG k- ε 2d d PBS. SST k- ω

4500 4000

Nu

6000

: Exp. : C a lc . S p a la rt-A . R N G k−ε S S T k−ω

3500 3000 2500 2000

4000

Nu

1500 1000 500 -1 .0

-0 .5

0 .0

0 .5

1 .0

1 .5 5

2 .0

S /C x 2000

Fig. 13 Comparison n of the local Nu N over the bllade mid span ussing three differrent turbulencee 3D models an nd DBS. -1.0

-0.5

0.0

0.5

1.0

1.5

2..0

S//Cx

Fig. 10 Comparison of thee local Nu overr the blade miid span using threee different turrbulence 2D mo odels and PBS.

Figss. 14-15 show local Nu num mber contours over the blade span s for the 2D 2 and 3D models m computted with three turbulence models m and with two ddifferent

28

Evalluation of Diffferent Turbule ence Models and a Numerica al Solvers for a Transonic Turbine Blade Cascade

numerical sollvers PBS andd DBS ) in co omparison witth the experimeental and anoother numericcal data. Thesse distribution were w obtained with 100% Reeex Case and P Pr = 0.7. The DBS as general giives a good agreement a witth another experrimental and numerical daata as shown iin Fig. 11, Fig. 13 1 and Fig. 15. The Spalart-A Allmaras modeel (ee) Nu contours for SST k-ω model m and 3D-PB BS Fig. 144 Nu contourrs over the blade span usin ng three differen nt turbulence 3D Models and PBS. P

(a) Ref. [1] loccal Nu contourss of experimenttal data over th he blade Span

(a) Nu u contours for Spalart-Allmar S ras model and 33D-DBS

(b) Ref. [1] loocal Nu contou urs of numerica al data over th he blade Span

(b b) Nu contours for RNG k-Є model m and 3D-D DBS

(c) Nu contou urs for Spalart--Allmaras mod del and 3D-PBS S

(cc) Nu contours for SST k-ω model m and 3D-DBS Fig. 155 Nu contourrs over the blade span usin ng three differen nt turbulence 3D Models and DBS. D

(d) Nu coontours for RNG G k-Є model an nd 3D-PBS

gives a very good agreement ovver the turbinne blade span with w the anotther validatedd data and tthe best

Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

agreement shown over the leading edge and the suction side as shown in Fig. 11, Fig. 13 and Fig. 15. The other turbulence models with the DBS give a good agreement over the turbine blade span and the best agreement with RNG k-Є and SST k-ω models shown on the pressure and suction sides and bad agreement near the leading edge due to transition relaminarization effect which occur near the leading edge. Tables 3-4 present the average Nu over the blade mid span and whole span percentage of uncertainty comparison between every turbulence model used in the Table 3 Average Nu % of deviation over the turbine blade mid span for 2D cases. 2D PBS 2D DBS

Spalart-Allmaras 4.7 4.39

RNG k-Є 27.54 9.56

SST k-ω 7.638 4.42

Table 4 Average Nu % of deviation over the turbine blade span for 3D cases. 3D PBS 3D DBS

Spalart-Allmaras 5.49 4.2

RNG k-Є 28.38 13.76

SST k-ω 6.95 4.7

29

present study with PBS and DBS for the 2D and 3D models. From Tables 3-4 concluded that generally the DBP gives more accurate results than the PBS for each turbulence model and for any shape of grid (2D and 3D models) and the Spalart-Allmaras turbulent model gives more accurate average Nu over the blade span than the other turbulent models for the 3D model so the Spalart-Allmaras model chosen to predict the heat transfer over a turbine blade for a different loads. The horse shoe vortex shown over the suction sides near the unheated end wall (T = To in) and covers the suction side from the geometric throat to the trailing edge along the uncovered portion of the suction side as shown in Fig. 16(b). No vortex shown over the whole pressure side as shown in Fig. 16(a).

8. Conclusions • Splart-Allmaras model with the DBS gives the best agreement of the calculated Nu with previous experimental results for both 2D and 3D models. In addition DBS gives more accurate results than the PBS for all turbulence models which was used in the present study; • No vortex appeared over the pressure side but a horse shoe vortex appeared on the suction side;

(a) Pressure Side and leading edge

• Three peaks of Nu appeared in the present investigation, the known peak near the stagnation and other two peaks near the end walls. This conclusion indicate the need to use 3D models for studding the heat transfer over the turbine blade due to the complex structure of secondary flows.

References [1]

[2] (b) Suction side Fig. 16 Stream lines around the half span turbine blade cascade.

[3]

P.W. Giel , R.J. Boyle, R.S. Bunker , Measurements and predictions of heat transfer on rotor blades in a transonic turbine cascade, ASME J. of Turbomachinery 126 (1) (2004) 110-121. H.H. Cho, D.H. Rhee, Effect of vane/blade relative position on heat transfer characteristics in a stationary turbine blade: Part 2. Blade surface, International Journal of Thermal Sciences 47 (11) (2008) 1544-1554. J.C. Han, J.Y Ahn, M.T. Schobeiri, H.K. Moon, Effect of

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Evaluation of Different Turbulence Models and Numerical Solvers for a Transonic Turbine Blade Cascade

rotation on leading edge region film cooling of a gas turbine blade with three rows of film cooling holes, International Journal of Heat and Mass Transfer 50 (2007) 15-25. [4] M.H. Albeirutty, A.S. Alghamdi, Y.S. Najjar , Heat transfer analysis for a multistage gas turbine using different blade-cooling schemes, Applied Thermal Engineering 24 (4) (2004) 563-577. [5] S.V. Ekkad, Y.P. Lu, D. Allison, Turbine blade showerhead film cooling: Influence of hole angle and shaping, International Journal of Heat and Fluid Flow 28 (5) (2007) 922-931. [6] V.K. Garg, Heat transfer on a film-cooled rotating blade, International Journal of Heat and Fluid Flow 21 (2) (2000) 134-145. [7] J.M. Miao, H.K. Ching, Numerical simulation of film-cooling concave plate as coolant jet passes through two rows of holes with various orientations of coolant flow, International Journal of Heat and Mass Transfer 49 (2006) 557-574. [8] A.A. Ameri, K. Ajmani, Evaluation of predicted heat transfer on a transonic blade using ν 2-ƒ models, ASME Turbo Expo 2004: Power for Land, Sea, and Air, June 2004. [9] V.K. Garg, A.A. Ameri, Two-equation turbulence models for prediction of heat transfer on a transonic turbine blade, International Journal of Heat and Fluid Flow 22 (6) (2001) 593-602. [10] P.W. Giel, S. Djouimaa, L. Messaoudi, Transonic turbine blade loading calculations using different turbulence models-effects of reflecting and non-reflecting boundary

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