Hybrid RANS-LES methods offer a means of reducing computational cost and setup time to simulate transitional flows. Several methods are evaluated in ANSYS CFX, including Scale-Adaptive Simulation (SAS), Shielded Detached Eddy Simulation (SDES), Stress-Blended Eddy Simulation (SBES), and Zonal Large Eddy Simulation (ZLES), along with a no-model laminar simulation. Each is used to simulate an adiabatic flat plate film cooling experiment of a shaped hole at low Reynolds number. Adiabatic effectiveness is calculated for Blowing Ratio (BR) = 1.5 and Density Ratio (DR) = 1.5. The ZLES method and laminar simulation most accurately match experimental lateral-average adiabatic effectiveness along the streamwise direction from the trailing edge of the hole to 35 hole diameters downstream of the hole (X/D = 0 to X/D = 35), with RMS deviations of 5.1% and 4.2%, and maximum deviations of 8% and 11%, respectively. The accuracy of these models is attributed to the resolution of turbulent structures in not only the mixing region but in the upstream boundary layer as well, where the other methods utilize RANS and do not switch to LES.
Modern gas turbine engine performance is driven through technologies enabling higher turbine inlet temperatures. One critical enabling technology for increasing turbine temperatures is the ability to cool the surfaces of rotating and stationary components in the turbine section by injecting a film of lower temperature air over the surface to shield it from the extreme gas temperatures exiting the combustor. With the advantages of film cooling in gas turbine engines, there comes the added complexity of the fluid dynamics and heat transfer involved. Film cooling flows are complex and boundary-layer dominated, and injecting film cooling into high-pressure turbine sections requires extensive computational resources to accurately predict.
Substantial research has been invested in better understanding and optimizing film cooling flows [
Subgrid-scale turbulence models are needed when small-scale physics, smaller than what can be resolved by the computational mesh, are important to the overall solution [
The hybrid models are evaluated in their ability to simulate the experimental film cooling flow of a flat plate with a shaped hole at low Reynolds number, with a focus on their ability to predict the adiabatic effectiveness. The relative advantages and weaknesses of each model in predicting such film cooling flow are then apparent, and a preferred hybrid model for predicting low Reynolds number film cooling is identified. The added understanding of the benefits and shortcomings of hybrid models is important in the design of industrial applications of film cooling, where the designer may prefer a unified model to reduce setup time when predicting the effectiveness of a particular cooling scheme. This work also bears importance to research and academia, as multiple models in development have seen limited evaluation in the application of film cooling. The rapidly-growing development of turbulence models offers many options, and this serves to catalogue the most recent of those models. Novelty is in the comparison of each model for film cooling and in the overall accuracy attained for the particular shaped hole film cooling case. This will illuminate existing literature by offering the results of each model for comparison and by describing the methodology used to obtain the results.
The Navier-Stokes momentum equation [
ρ ∂ u i ∂ t + ρ ∂ ∂ x j ( u j u i ) = − ∂ p ∂ x i + ∂ ∂ x j ( 2 μ S i j ) (1)
ρ ∂ u ¯ i ∂ t + ρ ∂ u ¯ i u ¯ j ∂ x j = − ∂ p ¯ ∂ x i + ∂ ∂ x j ( 2 μ S ¯ i j − ρ u ′ i u ′ j ¯ ) (2)
Here, u is the velocity vector, x is the direction, t is time, p is pressure, ρ is density, μ is dynamic viscosity, and S is the strain rate tensor, and the overbar represents the mean-flow property.
The Boussinesq eddy viscosity approximation [
− ρ u ′ i u ′ j ¯ = μ t ( ∂ u i ¯ ∂ x j + ∂ u j ¯ ∂ x i ) − 2 3 ρ k δ i j (3)
where μ t is the turbulent eddy viscosity, k is the turbulent kinetic energy, and δ i j is the Kronecker delta.
In order to calculate μ t , the k-ω model developed by Wilcox [
∂ ( ρ k ) ∂ t + ∂ ( ρ u j k ) ∂ x j = P k − β k ρ ω k + ∂ ∂ x j [ ( μ + σ k ρ k ω ) ∂ k ∂ x j ] (4)
∂ ( ρ ω ) ∂ t + ∂ ( ρ u j ω ) ∂ x j = α ω k P k − β ω ρ ω 2 + ∂ ∂ x j [ ( μ + σ ω ρ k ω ) ∂ ω ∂ x j ] + p σ d ω ∂ k ∂ x j ∂ ω ∂ x j (5)
where P k = τ i j ∂ u i ∂ x j , and τ i j = μ t ( 2 S i j − 2 3 ∂ u k ∂ x k δ i j ) − 2 3 ρ k δ i j .
Additionally, β k = 9 / 100 , σ k = 3 / 5 , α = 13 / 25 , σ ω = 1 / 2 , and σ d = 1 / 8 if ∂ k ∂ x j ∂ ω ∂ x j > 0 , else σ d = 0 . Also, β ω = 0.0708 1 + 85 χ ω 1 − 100 χ ω , where χ ω = Ω i j Ω j k S k i ( β k ω ) 3 , and where Ω is the mean rotation tensor.
The shear stress transport (SST) model, developed by Menter [
μ t = ρ k ω 1 max [ 1 α * , S F 2 0.31 ω ] (6)
where α * = 0.024 + R e t / 6 1 + R e t / 6 , F 2 = tanh ( ϕ 2 2 ) , and ϕ 2 = max [ 2 k 0.09 ω d , 500 μ ρ d 2 ω ] . Additionally, R e t = ρ k μ ω , and d is the distance to the nearest surface.
This SST model is used in the RANS regions of all the hybrid models as well as in the URANS model.
Large eddy simulation, first proposed by Smagorinsky [
∂ ∂ t ( ρ u ¯ i ) + ∂ ∂ x j ( ρ u ¯ i u ¯ j ) = − ∂ p ¯ ∂ x i + ∂ ∂ x j ( μ ∂ u ¯ i ∂ x j ) − ∂ τ i j ∂ x j (7)
where τ i j defines the subgrid-scale stress:
τ i j ≡ ρ u i u j ¯ − ρ u ¯ i u ¯ j (8)
The sub-grid scale model then models the deviatoric part of τ i j , τ i j − 1 3 τ k k δ i j . The Smagorinsky model does this with:
τ i j − 1 3 τ k k δ i j = 2 μ t S i j (9)
where μ t is the turbulent eddy viscosity, modeled in the Smagorinsky model by:
μ t = ρ ( C S Δ ) 2 | S ¯ | (10)
where | S ¯ | is defined by | S ¯ | ≡ 2 S ¯ i j S ¯ i j , C S is the Smagorinsky constant which typically ranges from 0.1 to 0.2, and Δ is the local grid scale, computed as the cube root of the element volume.
The wall-adapting local eddy-viscosity (WALE) model [
μ t = ρ ( C w Δ ) 2 ( S i j d S i j d ) 3 / 2 ( S ¯ i j S ¯ i j ) 5 / 2 + ( S i j d S i j d ) 5 / 4 (11)
where S i j d is defined by:
S i j d = 1 2 ( g ¯ i j 2 + g ¯ j i 2 ) − 1 3 δ i j g ¯ k k 2 (12)
where g is velocity gradient tensor, g ¯ i j = ∂ u ¯ i ∂ x j . Additionally, C w is the Smagorinsky constant which is set to 0.5. This LES WALE model is used in the LES region of the Zonal LES method.
Four different hybrid models are evaluated: scale adaptive simulation, shielded detached eddy simulation, stress-blended eddy simulation, and zonal eddy simulation. One variation of the zonal eddy simulation is examined, along with the URANS result and a no-model laminar result. The RANS model used in the hybrid methods as well as the URANS model is the k-ω shear stress transport model [
A brief description of each model is given. Additional details of each model can be found in the referenced sources.
The scale-adaptive simulation model is an improved URANS formulation which dynamically adjusts from the URANS modeled turbulence to resolved structures using the introduction of a von Kármán length scale in the turbulence scale equation. This results in an LES-like behavior in unsteady regions of the flow and allowing resolution of the turbulent spectrum, while maintaining the RANS in the rest of the domain. The model was first described by Menter and Egorov [
The SAS model modifies the RANS-SST turbulent eddy frequency transport equation, shown in Equation (5), by the addition of a source term, Q S A S , to the RHS of the ω transport equation. This source term is computed using the von Kármán constant, κ :
Q S A S = max [ ρ ζ 2 κ S 1 2 ( L L v K ) 2 − C 2 ρ k σ ϕ max ( 1 ω 2 ∂ ω ∂ x j ∂ ω ∂ x j , 1 k 2 ∂ k ∂ x j ∂ k ∂ x j ) , 0 ] (13)
where ζ 2 = 3.51 , C = 2 , σ ϕ = 2 / 3 , L is the modeled turbulence length scale and L v K is the von Kármán length-scale:
L = k / ( c μ 1 / 4 ω ) (14)
L v K = κ s | U ″ | (15)
where c μ is a replacement in the k transport equation for β k , S 1 is the scalar invariant of the strain rate tensor, and U ″ is the second velocity derivative.
Detached eddy simulation (DES) was the first proposed hybrid approach combining LES and RANS. The concept behind DES is to maintain RANS modeling near the wall where eddies are attached, but to switch to LES where the flow separates and the eddies become detached. The DES model replaces the distance term d in the RANS model with a distance function d ˜ :
d ˜ = min ( d , C D E S Δ ) (16)
where C D E S is an empirical constant with a value of 0.65.
To incorporate DES into the k-ω SST model, the dissipation term in the turbulent kinetic energy transport equation, shown in Equation (4), is modified to include the function F D E S :
ε k = ρ β k k ω F D E S (17)
where F D E S is defined by the turbulence length scale, L t , the maximum grid length in any direction, Δ max , and C D E S :
F D E S = max ( L t C D E S Δ max , 1 ) (18)
L t = k β k ω (19)
In order to prevent this limiter from becoming active in the attached portion of the boundary layer which can happen when Δ max is less than the boundary layer thickness, causing a grid-induced separation, the delayed DES model (DDES) [
F D E S = max [ L t C D E S Δ max ( 1 − f D D E S ) , 1 ] (20)
f D D E S = 1 − tanh [ ( C d 1 r d ) C d 2 ] (21)
where C d 1 = 20 , C d 2 = 3 , and r d is computed by:
r d = μ t + μ ρ κ 2 d 2 0.5 ( S 2 + Ω 2 ) (22)
in which S is the magnitude of the strain rate tensor, Ω is the magnitude of the vorticity tensor.
Improving upon DES, the improved delayed DES (IDDES) model [
Δ S D E S = max ( Volume 3 , 0.2 Δ max ) (23)
This SDES model, described by Gritskevich, Garbaruk, Schütze, and Menter [
An even more recent development is the stress-blended eddy simulation (SBES), which is recommended to supersede SDES. It employs the same shielding function as SDES, but also adds the ability to blend between the RANS and LES regions. The modeled stress tensor of the RANS and LES regions is blended:
τ i j S B E S = f S D E S τ i j R A N S + ( 1 − f S D E S ) τ i j L E S (24)
This model has demonstrated faster development of turbulent scales over SDES and has predicted a better match of data in turbulent mixing problems. The SBES model was developed and described by Menter [
The last hybrid method studied is zonal LES (ZLES). In this model an LES zone is specified by the user. An interface is created at the boundary of the zone, where the modelled unresolved turbulence of the RANS model is converted to resolved fluctuations for the LES zone. To do this, a source term is added to the momentum equations and a sink term is added to the modelled turbulent kinetic energy equation:
F m o m , i = ρ u f , i Δ t (25)
F k = − 0.5 ρ u f , i 2 Δ t (26)
where Δ t is the timestep size and u f , i is the velocity fluctuation to be transferred. The velocity fluctuation is determined by a random flow generator that uses the RANS turbulent length and time scales for input:
u f , i = 2 3 k 2 N ∑ n = 1 N [ p i n cos ( a r g n ) + q i n sin ( a r g n ) ] (27)
where:
p i n = ε i j k η j n d k n , q i n = ε i j k ξ j n d k n (28)
a r g n = 2 π ( d i n x i L t + ω n t τ t ) (29)
where the length scale is determined by L t = 0.5 k / c μ ω , and the time scale is τ t = L t / k . Additionally, random distributions N ( ϕ , ψ ) , where ϕ is the mean and ψ is the standard deviation, are the following:
η i n = N ( 0 , 1 ) , ξ i n = N ( 0 , 1 ) , d i n = N ( 0 , 0.5 ) , ω i n = N ( 1 , 1 ) (30)
This interfacing method for zonal LES is described by Menter, Garbaruk, and Smirinov [
To explore the impact of the Smagorinsky constant on the LES solution, the ZLES model is run again with this constant set to zero instead of using the WALE value. Setting the Smagorinsky constant to zero fixes the eddy viscosity to zero, effectively eliminating the subgrid-scale model. This would be analogous to DNS; however, the RANS zones are still specified, the LES advection scheme is still being used, and near-wall treatment is still applied. This model is referred to herein as Smag0 ZLES.
In addition to the hybrid LES methods, an unsteady no-model case is explored. No turbulence model or sub-grid scale model are used; only the laminar equations are employed. This can be considered an implicitly filtered LES as the grid acts as a low pass filter. This can also be referred to as a coarse or under-resolved DNS.
All the models studied are executed on the same mesh, with the same boundary conditions, initial conditions, timestep size, and time duration. The differences in the results are wholly due to the differences in the models, allowing examination of the similarities and differences amongst them and their relative utility for film cooling CFD. A brief description of each model is given in
The CFD model is based on the flat plate film cooling experiments conducted by Schroeder and Thole [
The cooling holes are a baseline fan-shape geometry that is publicly defined as the “7-7-7” shaped hole [
| Model | Description |
|---|---|
| Laminar | Laminar equations only, no turbulence model Less time per iteration due to fewer equations |
| SST | Modified k-ω URANS, no subgrid-scale modeling |
| SAS | Hybrid but no subgrid-scale model Dynamically resolves structures based on von Kármán length scale Most stable scale-resolving model, but can remain in URANS for mildly separated boundary layers and undisturbed channels with lower instability |
| SDES | Hybrid with subgrid-scale model Shielding (blending) between zones Grid-induced separation mitigated, but possible Asymptotic shielding, preserves νt. Requires strong instability to activate LES Superseded by SBES |
| SBES | Hybrid with subgrid-scale model Improved shielding (blending) between zones includes shear stress term Faster transition to LES in separating shear layers Relatively new, limited published research |
| Zonal LES | Hybrid with subgrid-scale model in explicitly defined zone Transition from modeled to resolved turbulence RANS and LES zones must be defined by user Synthetic turbulence model required at RANS-LES interface |
the 30˚ injection angle and assume symmetry. The geometry of the hole is shown in
| Injection angle | 30˚ |
|---|---|
| Lm/D | 2.5 |
| Llat/D | 3.5 |
| L/D | 6 |
| Layback angle, βfwd | 7˚ |
| Lateral angle, βlat | 7˚ |
| Pitch/Diameter | 6 |
| Area ratio | 2.5 |
This hole shape provides challenges in resolving the inherit unsteadiness of the flows, making it well-suited to for this study.
The mesh used in this study is a structured grid generated in the commercial CFD mesher ANSYS ICEM. The hole is meshed using an O-grid structure, and the test section and plenum are block-structured. The mesh was initially generated for half the domain then mirrored to guarantee symmetry and rule out any possible mesh-induced flow asymmetry. The first layer thickness of the cooled wall obtained a nominal y+ of 3, and a nominal x+ and z+ of 12. A y+ less than one was desirable but traded off for maintaining a low aspect ratio and reducing x+ and z+ to ensure minimal filter shape biasing of the flow within the boundary layer. The higher y+ is mitigated by this being only an adiabatic mixing problem with scalable wall functions that are intrinsic to the k-ω SST model in CFX. The final mesh size was 18 million elements. The side view is shown with a closeup near the hole in
| x-direction | y-direction | z-direction | time | |
|---|---|---|---|---|
| Wall-Modeled LES Requirement (CFX) [ | Nx = 10 | Ny = 30 to 40 | Nz = 20 | CFL ≈ 1 |
| x+ = 40 | Ny = 30 | z+ = 20 | CFL ≈ 1 | |
| Wall-Resolved LES Requirement (CFX) [ | dx ≤ LKol | dy ≤ LKol | dz ≤ LKol | dt ≤ tKol |
| x+ = 12 | ||||
| DNS Requirement [ | x+ = 12, dx < LKol | Ny = 48, y+ = 3 ± 1 (first point in viscous sublayer), dy < LKol | Nz = 120, z+ = 12, dz < LKol | CFL < 2, dt < 10tKol |
are resolved well beyond the inertial subrange for the relatively low Reynolds number flow.
Having established that the mesh meets notional wall-resolved LES quality requirements, a grid sensitivity study is subsequently performed by applying varying refinement levels to the grid. The levels are incremented in quarter increments of the base length of the original mesh, from 1/4 to 5/4, and the timestep size is held constant. Although grid convergence for LES is the DNS solution, which is outside the scope and computational resources for this work, stability in the simulated flow field can be observed as the mesh is refined, suggesting grid insensitivity is achieved by the original mesh. In
an RMS of 1.76% from the 3/4 to the 1/1 level and an RMS of 0.68% from the 1/1 to 5/4 level grid. This demonstrates that the velocity upstream of the hole is grid insensitive for the 1/1 level grid used.
To observe the convergence of the adiabatic effectiveness on the cooled surface,
Ideal gas properties for air are used with Sutherland’s formula for dynamic viscosity and thermal conductivity is applied with a reference Prandtl number for of 0.707 at 293.15 K and 1 atm. The numerical equations for total energy including viscous work, momentum and mass, wall scale, and the applicable turbulence model were iterated in the solver. For all of hybrid models, the 2nd order high-resolution advection scheme, developed by Barth and Jespersen [
To initialize each of the hybrid simulations, a steady state RANS solution was first attained, which was then used to initialize a large timestep size (1e−3 s) URANS simulation which ran for one complete flow-through. That was then
used to initialize a to a smaller timestep size (1e−4 s) URANS simulation which ran for three more flow-throughs. The length of the domain from inlet to outlet is 0.424 m, and the nominal flow velocity is 10 m/s, so 424 timesteps were required for one flow-through at this timestep size. The URANS results presented are of the smaller timestep simulation, with data collection starting after the first smaller timestep flow-through to the end of the fourth flow-through, with the time-average results of this range exhibiting good convergence (<5% change between flow-throughs in centerline and lateral adiabatic effectiveness along the cooled plate). The final state of the URANS solution was then used to initialize each of the hybrid simulations. The hybrid simulations used a timestep size of 1e−4 s as well and were run for one complete flow-through before data sampling, and then run for five more flow-throughs, collecting the time-averaged data presented herein. The time-averaged results were well converged as shown later, and the timestep size was sufficient in resolving the inertial subrange frequencies.
The inlet was placed 12D upstream of the hole and set to a velocity inlet at a temperature of 295 K. The experimental 2D velocity and turbulence intensity profiles measured 2.3D upstream of the hole were applied to the inlet without adjustment. The measured turbulence intensity was 0.5% across most of the test section and increases to 13% near the wall. The turbulence length scale was set to 22% of the inlet boundary layer thickness. The cooling plenum inlet is located 5D from the hole entrance. The coolant flow is set to a temperature of 196.7 K to accomplish a density ratio of 1.5 and the mass flow rate is set to achieve a blowing ratio of 1.5. The turbulence intensity at the plenum inlet is set to 5%, and turbulent viscosity ratio is set to 10. The outlet was located 40D downstream of the hole and is set to a static pressure opening at 1 atm, with backflow enabled. The test section height was set to 5D, with a symmetry condition applied to the far wall, understanding the value of reducing the domain size versus the possible source for inaccuracy since the entire test section height is not modelled. The lateral sides of the domain were set to translational periodic interfaces. The remaining boundaries were set to no-slip adiabatic walls. A diagram of the computational domain is provided in
The time-averaged adiabatic effectiveness contours on the cooled surface are presented in
| Geometry | Hole diameter | 7.75 mm |
|---|---|---|
| Inlet distance from hole center | 12D | |
| Plenum height | 5D | |
| Outlet distance from hole center | 40D |
| Mesh | Type | Symmetric multi-block structured with O-grid |
|---|---|---|
| Size | 18M Elements | |
| First layer dimensions | x+, y+, z+ = 12, 3, 12 | |
| Domain | Gas properties | Air ideal gas with Sutherland’s, Pr = 0.707 |
| Time duration | 6 Flow-throughs | |
| Timestep size | 1e−4 s | |
| Timesteps per flow-through | 424 | |
| RANS turbulence model | k-ω shear stress transport | |
| Boundary conditions | Main inlet | Experimental velocity and turbulence profile |
| Plenum inlet | Mass flow inlet | |
| Outlet | Static pressure opening with backflow | |
| Sides | Translational periodic | |
| Far wall | Symmetry | |
| All other walls | Adiabatic no-slip | |
| Solver settings | Advection scheme | RANS: high resolution, LES: central differencing |
| Turbulence numerics | 2nd Order high resolution | |
| Transient scheme | 2nd Order backward Euler, 10 inner iterations |
the other hybrid models, the SBES, SDES, and SAS models exhibit an asymmetry where the streak favors one side or the other that is not apparent to the same extent in the ZLES model or in the test data. Albeit, the ZLES model and test data are not perfectly symmetric, and symmetry is not necessary for this complex flow field. Furthermore, the experiments of Aghasi and Gutmark [
To quantify the accuracy of each model in predicting film cooling effectiveness, the adiabatic effectiveness along the hole centerline and the lateral-average adiabatic effectiveness are plotted in
More important to the overall effectiveness of the cooling scheme, the lateral-average effectiveness on the surface is best captured by the ZLES and laminar
simulation. The largest absolute deviations from the test data for these are immediately downstream of the trailing edge of the hole, with a value of 8% for the ZLES and a value of 12% for the laminar simulation. The RMS of the lateral-average deviation is also lowest for the ZLES and laminar simulation, with values of 5% and 4%, respectively.
In order to verify that a sufficient number of flow-throughs have been run for transient averaging, the convergence of the centerline and lateral-average effectiveness after each complete flow-through is evaluated. For reference, plots of the centerline and lateral-average effectiveness convergence after each flow through of the ZLES model are shown in
In
The turbulence spectra for each model are presented in
fast Fourier transform of the time signal with a Hann filter. The spectra may be difficult to discern from one another, but it should be noted that the turbulence simulated by ZLES and Smag0 ZLES model spectra are well above those of the other models. It can also be seen that the −5/3 slope of the inertial range is only attained for a short range, roughly between 500 and 2000 Hz. The relatively short inertial subrange is consistent with low Reynolds number spectra [
In
In order to quantify the energy contained by the spectra of each model,
significantly higher overall turbulence spectrum at the hole leading edge. Furthermore, even though the laminar model carries no subgrid turbulence, it still is capturing a significantly larger κE than the SBES, SDES, and SAS models.
The lower turbulence calculated by the SBES and SDES models is attributed to the boundary layer shielding functions forcing a RANS zone upstream of the hole near the wall, which does not allow turbulence to develop upstream of the hole. This behavior of attached eddies until geometry-induced separation occurs is consistent with detached eddy simulation theory. Unfortunately, in this case it means that the near-wall turbulence upstream is not being sufficiently developed by either SBES or SDES. Additionally, despite the SAS model not having such shielding functions and having a larger scale-resolving zone upstream of the hole, the turbulence is not developing nearly as much as in the ZLES model or even the SDES and SBES.
A computational study of hybrid RANS-LES models was performed to examine the ability of modern hybrid models in predicting film cooling effectiveness of a low Reynolds number flat plate film cooling experiment. It was found that the shielding of the RANS boundary layer by the SBES and SDES models precluded the development of upstream turbulence that impacted the film cooling effectiveness. The SBES, SDES, and SAS models did not resolve a sufficient amount of turbulence and were rendered ineffective for low Reynolds film cooling simulation. Zonal LES and even the no-model laminar simulation exhibited higher upstream turbulence and were more accurate in capturing the film cooling effectiveness. Thus, despite the convenience of generic RANS-LES hybrid models, based on the results, they are not herein recommended for low Reynolds number film cooling. Further research is needed to examine their usefulness for higher Reynolds flow, closer to actual engine conditions, although the shielding functions will still act to shield the RANS boundary layer upstream of the cooling hole jet. Should a hybrid RANS-LES approach be desired, zonal LES is the recommended choice for low Reynolds film cooling applications by these conclusions.
This work could not have been accomplished without the support of Dr. Karen Thole and Turbine Heat Transfer and Aerodynamics Group at Penn State, as well as the resources made available by the Ohio Supercomputer Center, and funding provided by the Ohio Federal Research Network.
The authors declare no conflicts of interest regarding the publication of this paper.
Boehler, M., Sudesh, A. and Turner, M. (2020) Hybrid RANS-LES of Shaped Hole Film Cooling on an Adiabatic Flat Plate at Low Reynolds Number. Open Journal of Fluid Dynamics, 10, 317-341. https://doi.org/10.4236/ojfd.2020.104019
BR: blowing ratio, ρcuc/ρ∞u∞
D: hole diameter
DR: density ratio
E(κ): energy spectrum function
k: turbulent kinetic energy
Reθ: momentum thickness Reynolds number, θu∞/ν
Reτ: shear velocity Reynolds number, δuτ/ν
Tc: coolant temperature
uc: coolant jet velocity
uτ: shear velocity
x+: streamwise dimensionless wall distance, uτx/ν
y+: wall-normal dimensionless wall distance, uτy/ν
z+: spanwise dimensionless wall distance, uτz/ν
ε: turbulence dissipation rate
η: adiabatic effectiveness, T ∞ − T T ∞ − T c
κ: wavenumber