Application of a Ranque–Hilsch Vortex Flow in an Internal Cooling of a Gas Turbine Engine Blade

The life cycle of a gas turbine engine blade is most affected by higher operating temperatures, constant centripetal loading, and thermally induced stresses, particularly during start-up and shutdown. The thermal efficiency of the engine increases with the increase of the pressure ratio and the firing temperature, which increases the turbine rotor inlet temperature. As the firing temperature increases, the heat transferred to the turbine also increases, rising above the material temperature threshold, which requires mitigation measures of material failure if upholding the high temperature is desired to maintain high efficiency. This requires internal cooling of the rotor blades (Fig. 1). Significant research has been going on for decades to design an internal cooling system, particularly for the first-stage blades, to achieve higher firing temperatures. Effective internal cooling of the rotating blades is a significant challenge, compounded by wake-induced turbulence and unfavorable area ratios between inner and outer surfaces [1,2]. This can cause formidable challenges to turbine internal cooling.

Various cooling techniques are applied to the turbine blade to keep the working temperature within a safety limit [3–5]. It is a common practice to cool high-pressure turbine blades using air from the compressor, which is routed through the turbine blades, thereby lowering its temperature. Swirl cooling is one of the many techniques used for such cooling. The idea is to route air from the compressor through the turbine blade's internal passages forming a swirl flow [6]. The first rows of turbine blades typically operate at a temperature that exceeds 1200 °C [7–9] and, therefore, may greatly benefit from internal cooling if a higher firing temperature is to be entertained.

Analytical and experimental modeling of internal swirl cooling systems in the leading-edge area of the blade could result in the optimization of turbine blade designs concerning heat transfer, cost, performance, and downtime [10–12]. Adding complexity to the demanding task of managing high temperatures without its accompanying penalties is the driving desire for a long-term life cycle without frequent inspections and overhauls. Turbine blades withstand high temperatures and constant mechanical stresses, which limit the turbine blade life cycle and may also cause permanent material deformation [13,14]. These can also cause local stresses on the material while contributing to material creep [15]. Some innovative techniques have been proposed to improve the convective heat transfer for internal cooling of gas turbine airfoils, including rib turbulators, pin fins, dimpled surfaces, impingement cooling, and swirl flow cooling [16–20].

Of particular interest here, the topic of our research is “swirl cooling,” which induces a reverse flow. One such innovative approach is discussed by Glezer et al. [14] who present experimental results comparing three separate studies. The research provided a better understanding of the screw-shaped swirl cooling technique for heat transfer in internal swirl flow, where heated walls were applied and a screw-shaped cooling swirl was generated, introducing flow through discrete tangential slots. The authors mentioned that the Coriolis forces play an important role in enhancing the internal heat transfer when their direction coincides with a tangential velocity vector of the swirl flow.

Another article on the subject of swirl cooling [21] states that the local surface Nusselt number increases when increasing the Re number (the range in this study was from 6000 to about 20,000). As a result, the local swirl chamber heat transfer and flow structure are linked to increased advection as well as notable alterations to vortex behavior near the concave surfaces of the swirl chamber. One key result was that, along with the Nusselt number, the changes in surface heat transfer downstream of each inlet increased sharply when compared to other locations. Hedlund and Ligrani [21] observe that as the turbulent flow becomes more pronounced, the axial and circumferential velocities get larger and intensify the turning of the flow from each inlet.

Other studies show that the blade internal swirl flow cooling is effective and can afford long-term life to blades especially if employed in tandem with advanced blade alloys [22,23]. Experiments conducted by Ligrani [4], Moon et al. [1], and Glezer et al. [13] introduced an internal cooling structure as one way to attend to high-temperature management in gas turbine cooling.

The aforementioned summaries of recent developments in turbine blade internal cooling show advances in several fronts. However, the idea of applying the Ranque–Hilsch vortex flow for internal blade cooling has not been investigated as it is presented in this article. Thus, the focus of this study is to conclusively prove that sustained reverse flow with temperature drop can be produced, which allows the use of the cold stream for a gas turbine blade internal cooling. The vortex flow is injected tangentially through air inlets to induce vortices, as shown in Fig. 2.

The following assumptions were imposed on Eq. (1):

• $vr=0$ and $∂∂z=0$⁠, because there is no flow in the radial direction and/or variation of properties in the z-direction.

• $ρgr=0$⁠, because there is no body force, is no buoyancy, and is homogeneous.

• $∂∂θ=0$⁠, because circumferentially symmetric flow properties do not vary with $θ$⁠.

Overall high S_N results indicate proximity to the air inlet, which restricts the axial component of the flow.

The geometrical model for the vortex flow cooling system was developed using solidworks 2020. The simulation domain was 3D, with seven rectangular inlets with a dimension of 44.45 × 22.68 mm each and a single outlet. A coaxial connection was selected with an outlet having a diameter of 40 mm.

The computational model simulates the flow separation of the incoming pressurized air into hot and cold streams in the cylindrical chamber, as validated in the experiment [24,25]. Input parameters, pressure, and temperature were obtained from experimental data. All inlet temperatures remain at 302 K determined at Re = 14,000. The cold exit pressure was left as that of the ambient, i.e., 0 Pa. It is assumed that the system operates in ambient conditions at the sea level. The set of pressure values presented in Table 1 is to be used for the numerical simulation pressure inlet boundary conditions. The pressure values were recorded from the experimental setup. Seven pressure sensors are located at each inlet of the chamber as shown in Fig. 6.

The energy equations were activated to handle the temperature effect with the gravitational acceleration accounted for. The coupled pressure velocity scheme was chosen for equation discretization. Other than the turbulence kinetic energy special discretization, all the variables were set as a second-order upwind scheme. However, the underrelaxation factors were left as the default values.

The geometrical model is shown in Figs. 4(a) and 4(b), and the mesh structure is shown in Figs. 4(b) and 4(c). The right end of the chamber stayed closed.

The physical domain was discretized using the ansys 2021 r2 version with an element size of 5 mm. The cell zones are the fluid domain occupied by compressed air and the cylindrical camber. The fluid domain has an outer diameter of 80 mm with a length of 1100 mm, while the solid body is 102 mm × 134 mm × 1100 mm. The volume of the fluid domain is 5.5489 × 10⁶ mm³, which is about 57.88% of the volume of the chamber.

The numerical simulation was performed using ansys 2021 r2 package. A 3D computational domain fluent model setup employing a steady-state pressure-based solver with a standard k-ε turbulence model was applied with a standard wall function, which is the-law-of-the-wall for mean velocity yields [1]. The setup of the solver is pressure based with an absolute velocity formulation for the steady-state standard k-ε model. The gravitational acceleration was activated to account for the body force influence. The energy equation was used to register the temperature variations throughout the model. The k-ε turbulence model was selected because it more accurately estimates the spreading rate of both planar and round jets. In addition, the model can offer superior output for flows such as rotation, boundary layers under adverse pressure gradients, recirculation, and separation.

The pressure inlet boundary condition kept the temperature at 302 K with an absolute reference frame. It was assumed that the air was admitted normally to the inlet boundary of the chamber. The turbulence effect was specified as turbulence intensity and viscosity ratio at the default values. The outlet boundary conditions are set as a pressure outlet at 0 Pa gauge pressure normal to the outlet boundary. The operating pressure and temperature were at standard temperature and pressure, whereas the operating density was 0 kg/m³. Zero density is assumed as an initial value to account for the compressibility effects during the swirl flow simulation process and avoid runaway errors with convergence.

To derive the numerical equation of pressure from the combination of continuity and momentum, we preferred to use a coupled pressure–velocity scheme. Least-square cell-based, second-order pressure, second-order upwind density and momentum, first-order upwind turbulent kinetic energy and turbulent dissipation rate, and second-order upwind energy were used for the spatial discretization process. To stabilize the numerical simulation and give a faster convergence, pseudo-transient form of the implicit underrelaxation transient formulation was checked for the steady-state computational fluid dynamics (CFD) analysis. The pseudo-transient relaxation factors are 0.5, 0.5, 1, 0.75, 0.75, 1, and 0.75 for pressure, momentum, density, body force, turbulent kinetic energy, turbulent dissipation rate, turbulent viscosity, and energy, respectively. The convergence conditions for the residual equations of continuity, x, y, and z-velocity, energy, and kinetic energy dissipation k-ε are 0.001, 0.001, 0.001, 0.001, 0.0001, 0.001, and 0.001. The simulation solution process was initialized as a hybrid.

In Eq. (10), P_k represents the generation of turbulence kinetic energy due to the mean velocity gradient. P_b is the generation of turbulence kinetic energy due to buoyancy. Y_M represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate.

The target space of a CFD model is divided into a finite number of grids for numerical analysis. These randomly selected grids are not necessarily optimum. A grid independence test was performed in this case, based on which around 2 million elements were concluded to be optimum. Increasing beyond that magnitude will provide more optimum results but at a high computational cost. Table 2 shows average exit parameters for different mesh sizes.

Pressure and temperature contours were also mapped for different grid sizes, but not shown here. On the basis of the grid-independent test for changes along the length of the chamber, Fig. 5, we conclude that the grid is fine enough, and for grids greater than 1.2 million cells, the solution has not changed significantly, and thus, the solution is assumed to be grid independent.

The main piece of the lab setup for the vortex flow cooling experiment is the LaVision Stereo-PIV system, which includes a LaVision PC, two ImagerproX cameras, two Nd-YAG lasers, and a LaVision particle seeder as shown in Fig. 6. The cameras are mounted on a stand equipped with stepper motors, allowing them to travel freely along the chamber length. The fluid is seeded with olive oil particles that have diameters in the range of 1–3 μm and a specific gravity of 0.703. These oil tracer particles are chosen because they are small enough that they have little inertia, validating that the tracer particle motion best reflects the actual flow path. The seeding particles in the fluid distribute the laser light, which is captured by the video acquisition system. The Nd-YAG laser is a Pegasus PIV with a wavelength of 527 nm and a maximum energy of 20 MJper pulse. The Nd-YAG laser serves as the illumination source for the PIV system and is manipulated through the appropriate use of optical instruments to produce a laser sheet of 2 mm thickness on the chamber's bottom wall. This setup allows the illumination of planes parallel to the vertical axis. Two high-speed and high-resolution CCD cameras (Phantom v7.3. 800_600 pixels, 12 bit) capture images of the illuminated PIV particles at a rate of 100 frames per second. With that frame rate, 2000 images are acquired over a period of 10 s, similar to the measurement duration in the experiment conducted by others [13,14].

DaVis is utilized to collect PIV data, all data points are taken and collected in one file set, which exports time-average velocity and is postprocessed in DaVis to be transferred into matlab. matlab cleans up DaVis raw data and allows the calculation of crucial flow field variables. All velocity calculations are conducted in matlab, followed by a file structure that prepares data for visualization in tecplot 360. One advantage of using tecplot 360 is that it provides powerful flow visualization options. Some macros are created to automate visualization procedures. To create smooth transitions between data points, a data interpolation scheme was employed.

Before committing to the use of the cold stream of the Ranque–Hilsch flow, it would behoove us to first scrutinize its occurrence within the theoretical scope presented earlier, with an accompanying temperature drop—significant enough to cool gas turbine blades. Then we will produce model-based experimental validation for assertions and simulation results.

The Navier–Stokes, centrifugal force, and centrifugal velocity equations can be used to show the prevalence of the reverse flow in the cylindrical chamber. Once preliminary values were entered, the experimental results show a radial pressure drop of about 140 Pa, which from the gauge total pressure at the inlet of approximately 340 Pa, as clearly illustrated in Fig. 7.

CFD simulations can further lay evidence to predict the loadings and flow distributions of blade rows, including for end-wall regions. As a tool, CFD can produce valuable outputs, albeit with its limitations in predicting turbine heat transfer, mainly because of constraints in modeling turbulence and vortices, the uncertainty of boundary conditions, and the inherent flow unsteadiness in turbomachinery. Here, we simulate velocity, temperature, and pressure distributions to show if their profiles support the notion of the existence of a reverse flow and if the continuity equation is satisfied.

Velocity: The velocity streamlines for the entire fluid domain are presented in Fig. 8. The largest velocity value is noticed at inlet 7 with a value of 30.5 m/s. There exist three reversed flow cases at inlets 1, 2, and 3. The velocity is minimum near the end of the chamber.

Temperature and Pressure: The temperature contour, Fig. 9(a), indicates separated cold and hot streams. These temperatures and pressure show the existence of a reverse flow with a temperature drop. The pressure distribution along the cylindrical chamber is shown in Fig. 9(b). The highest pressure is observed in front of inlet seven reading 979 Pa. However, after air inlet seven, the pressure distribution shows a significant drop resulting in −31 Pa.

Continuity Equation: To confirm the mass is balanced, we assess the mass flowrate at some distance “z” downstream of the flow. The goal here is to plot the mass flowrate at the three assumed Re numbers to confirm that the mass flowrate going into the cylindrical chamber matches the total mass flowrate going out of the chamber as cold and hot streams. Figure 10 shows that the mass flowrate going into the chamber from air inlets 1 through 7 (z-distance 4.5 through 20) totals the sum of reversed and unreversed flows leaving the chamber, which proves that the reverse vortex flow takes place inside the cylindrical chamber.

The total mass flowrate going into and through the system is 0.0535 kg/s. This is also the sum of the flowrate through each inlet, which is again the same as the flowrate at the outlet, as shown in Table 3.

Two distinct validation procedures were employed: the use of thermochromic liquid crystal and a video photograph depicting the reverse flow within the vortex passage.

Figure 6 shows only the test section and instrumentations used. However, this section contains the general setup of the test including the primary and secondary plenum. The image in this section depicts accessories such as perforated plates, mesh heaters, honeycombs, and baffle plates and the direction of the compressed air source and the expanded airflow path(s) direction. The purpose is to show how the air enters the plenum, gets heated, and travels into the vortex chamber via the seven air inlets. The cylindrical chamber is made of clear acrylic material and painted with thin-layer chromatography (TLC) to allow visualization of color changes within the required time range. As the air enters the plenum, Fig. 11, it passes into a rectangular heating mesh, leading to rectangular cross-sectional air inlets with individual hydraulic diameters (D_H) of 0.011 m. These air inlets are connected to the principal vortex chamber so that one surface is tangent to the chamber's inner circumference. Once the heated air enters the vortex chamber via the seven air inlets and the heated air comes into contact with the TLC, a color change is noticed.

The surface area optics suitable for the spatial resolution are painted, first, the cylindrical chamber is coated with liquid crystal paint, followed by black paint. A coupon is built, following the same process for calibration purposes, coated with liquid crystal paint, and followed by black paint.

TLC Data Collection: In-house software is used to synchronize the entire liquid crystal experiment. The data are collected by continuous polling after the system is heat soaked to the required temperature. The data acquisition step size is 0.5 s and the video extended interface is set to a time interval of 0.2 s. The video file is converted to an audio video interleave file and imported into the “liquid crystal image analyzer.”

Two cameras are used to collect the color change of the TLC, as shown in Fig. 12. Contact resistance and temperature drop through the wall are determined experimentally from the calibration process, and green time is measured simultaneously. By using seven calibrated thermocouples equally separated across the length of the cylindrical chamber, the temperature of the air entering the cylindrical chamber is measured. All measurements are collected when the cylindrical chamber is at a steady state and when the heating mesh on the plenum reaches 35 °C.

Once the green time image is calculated, then the probe locations are associated with columns of temperature readings in the temperature probe data file. Noise is edited out with an eraser image mask, followed by the region of interest definition using a polygon mask. These final images are then exported as joint photographic experts group (JPEG) image files. Cameras recording in DV format are set up to view the liquid crystal-coated surface of the test article, capturing 10 samples per second. The airflow rate is set as the liquid crystal transitions at 35 °C and regulator pressure at 6.89 KPa. Once the system is heat soaked, cameras start recording. When the paint has fully transitioned to blue, the cameras and data acquisition system are stopped manually.

The data points were taken at 33 locations at a distance of 19.81 mm from each other, as shown in Figs. 13 and 14. Data were collected at both locations, in between air inlets and in the middle of the air inlets (i.e., data point #2 and data point #4, respectively) to show the complicated flow and its variation.

Axial Velocity Distribution: As the Re number increases, the axial velocity, V_Z, intensifies, measuring higher on the outer wall region, as shown in Fig. 15. From upstream to downstream, the V_Z ranges from −2 to 7 m/s in all three Re numbers, as is expected in a cylindrical chamber. The outer wall region velocity increases across the length, reaching a maximum velocity at the second half of the vortex chamber. Another observation is the high V_Z in the direction of the outer wall region. Between the core and the chamber wall in the outer wall region, an inertia-driven vortex was observed, and a flow field pattern is measured that is critically different between high to low Re numbers.

Critical Swirl Number (S_N): The swirl number, the ratio of the axial flux of swirling momentum to that of the axial momentum, accounts for the nonsolid-body rotation, i.e., clues, not necessarily conclusively, the presence of a reverse flow. The critical swirl numbers at three Re numbers (Re = 7000, 14,000, and 21,000) are presented in Fig. 16. The data were recorded at 33 points in 19.81 mm distance from each other throughout the length of the swirl chamber. Figure 13 presents the locations of the data points taken along the length of the swirl chamber during the experiment. All measurements are collected in the center of the air inlets and in between air inlets.

Figure 16 presents S_N values that are as high as 3.5, indicating the high nondimensional circumferential moment of momentum relative to nondimensional axial momentum. The S_N magnitudes decrease steadily along the length of the chamber and then become almost horizontal. The highest S_N peak at z/D = 1.5 was measured as 3.5 during the experimental tests. The experiment for the chamber showed the existence of a reverse swirl flow scenario at all three Re numbers. At a Re = 14,000, a reverse swirl flow covers about 33% of the cylindrical chamber diameter. Distribution of the critical swirl number supports the existence of blade internal heat transfer with the reverse swirl flow condition.

Reverse Vortex Flow at the Center of the Cylindrical Chamber: The unexpected consequence of the vortex flow behavior needed more analysis to prove the occurrence of a reverse flow scenario. We launched a 3D stereo-PIV platform at the nominal Re = 13,639. Three cross-sectional areas were chosen at z/D = 3, 6, and 13 for this observation. The images were carefully studied looking for evidence of the reverse flow, and as such, a reverse flow was noticed in all the three cross-sectional areas. Stereo-PIV is a powerful tool to capture in detail how each droplet behaves and to map the axial velocity flow field. The velocity profile was captured for the Ranque–Hilsch vortex tube. The contour obtained from the experiment, shown in Fig. 15, proves the existence of a reverse flow inside the chamber of the Ranque–Hilsch vortex tube. The color-coded velocity profile represents the magnitude and the direction of flow for the swirled fluid. For the larger Reynolds number regime, i.e., closer to the wall passed the fifth inlet, we note that the fluid accelerates to the outlet with a local velocity of 7 m/s. However, looking at the center of the Ranque–Hilsch vortex tube at the same Reynolds number, the fluid velocity (shown with blue line in Fig. 16) has a magnitude of about 2 m/s in the reverse direction. When the flow becomes less turbulent, the velocity drops as it approaches the wall. Moreover, the flow direction adjusts to a stream-wise direction as the Reynolds number is reduced. The common feature for all the scenarios at the three Reynolds numbers is the existence of reverse flow at the center of the vortex tube.

Figure 17 presents the normalized axial velocity against the cylindrical chamber length, downstream of the flow. The pressure decreases downstream on the outer chamber wall, and the reverse flow at the centerline is increasingly visible.

As another validation of the model results, we decided to observe the flow pattern across the diameter of the chamber using video photography. For this purpose, we prepared a wire with a piece of string and a fabric attached at the end of the wire. The wire is then inserted at three different locations to see the direction of motion. The string moved in the direction of the flow in all three locations clearly, demonstrating the presence of a reverse flow as shown in the yellow arrows of Fig. 18. The string changes direction as it is moved across the internal boundary of the reversed flow.

Figures 19 and 20 show a comparison of temperature and pressure distribution in the vortex chamber as obtained from the experiment. The results look like a horizontal straight line for simulation performed at measured inlet pressure. Even though a minor temperature drop occurred in the chamber, the experimentally collected data points are nearly invisible for the existing working conditions of the vortex chamber. However, in CFD when the incoming air is increasingly compressed, the peak in temperature and pressure is more noticeable and delivers a wave-like distribution. One observation is the moving wave obtained its peak near the air inlets for both temperature and pressure, as shown in Figs 19 and 20, which is predictable. The temperature and pressure values have a noticeable slight increase as the vortex flow exits the chamber.

Figure 21 shows the temperature drop in the chamber at different elevated inlet pressure. As discussed in Fig. 19(b), the drop-in temperature in the chamber at low inlet pressure is very small and negligible. Nonetheless, when the inlet total pressure increases the drop of temperature will become very interesting. As shown in Fig. 21, there was a temperature drop of about 26 °C for an inlet total pressure of 50 kPa. The temperature drop increases as inlet pressure increases. However, increasing the inlet pressure beyond 300 kPa will not provide a further temperature drop. The CFD simulation was performed at an inlet pressure of 500 and 1000 kPa inlet pressure. The result overlaps with 300 kPa. Therefore, 300 kPa compression is the optimum inlet pressure to get a temperature drop of 67 °C in the chamber.

The air becomes more relaxed in the tube than in the chamber as shown in Fig. 22. At low inlet pressure, the expansion process is almost the same and linear compared to very high-pressure inlet.

The impact of increased firing temperature on the efficiency of a gas turbine engine is very well known and needs no detailed treatment here. Thus, the following is a brief outline of the subject to underline the possible flow extraction scenarios, which will have an impact on the Ranque–Hilsch vortex formation.

Equation (17) allows determining efficiency change as a function of $δa$ if extraction is after complete compression. The equation also accounts for the mixture temperature. The derivation for the mass extraction with full compression assumes isentropic relations and ideal gas.

The thermal efficiency of the gas turbine engine is shown in Table 4. The efficiency can be as high as 0.43 if the internal cooling of the blade allows an increase in gas temperature to 1800 K.

The primary objective of this research has been met by performing tests to prove the prevalence of a reverse flow in the swirl chamber that was designed to internally cool the gas turbine blade. The potential of the vortex tube, intended to produce vortex cooling without major casting changes, was demonstrated by using seven air inlets. The designed channel served as a Ranque-Hilsch vortex tube. The span of the outer-wall velocity increases due to the enlargement of the vortex flow core. As a result, the compressed air was separated into hot and cold streams in the swirling chamber with a potential temperature difference reaching over 250 °C.

Simulation results for the inlet pressures recorded from the experimental test confirmed that there is a drop-in temperature and pressure of the reversed flow. The cold exit temperature drop strongly depends on the level of compression for the inlet air, i.e., the point of air extraction from the compressor. Both the simulation and experimental results showed that the drop-in temperature and pressure is significant when the incoming air pressure increases. When the simulation experimented at 63 kPa and 100 kPa inlet pressure [26], the stream temperature cooled to 0 °C, i.e., yielding a temperature drop of about 29 °C. For the inlet pressure of 300 kPa, the temperature drop reaches 65 °C.

The TLC heat transfer test results exemplify how the Nu number was measured favorably at the middle length of the chamber, and these values decline downstream. The detailed flow behavior inside a Ranque–Hilsch vortex tube and flow reversal in the cylindrical swirl chamber share similarities. Based on this study, there was an impressive and conclusive presence of reverse axial flow at the core as illustrated in the experimental results. CFD simulation correlated the experimental results and validated the reverse flow with similar mass flowrates and pressure gradients of 0.0535 kg/s and 979 Pa, respectively. As part of the experiment, we determine the critical swirl number that has the potential to deliver the maximum axial velocity with the highest heat transfer at three different Reynolds numbers, 7,000, 14,000, and 21,000. The CFD established the reverse flow at Re = 14,000 and upheld the notion that higher pressure at the inlet can contribute to lowering the temperature at the exit from the chamber, favoring the cooling process. The thermal efficiency increases by about 3% when the blade is cooled by extracting 10% partially compressed air. This is a significant gain.

The concept was initiated by Asfaw Beyene who also supervised the research. Experiment was conducted by Daisy Galeana, and the simulation was done by Ashenafi Abebe. The paper was extracted from Daisy's thesis and edited by Asfaw Beyene.

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

The authors attest that all data for this study are included in the paper.

A =

area (m²)

D =

circular chamber diameter (m)

D_H =

hydraulic diameter of one swirl chamber inlet (m)

F =

centrifugal force (N)

g =

gravitational acceleration (m²/s)

H =

height (m)

k =

turbulent kinetic energy (m²/s²)

L =

cylindrical chamber length (m)

m =

mass (kg)

P =

pressure (Pa)

pps =

pints per second,=0.4732 l/s

Q =

mass flow rate (kg/s)

Re =

Reynolds number

r =

radial distance measured from chamber centerline (m)

STP =

standard temperature and pressure

S_N =

swirl number (critical)

V_z =

axial velocity (m/s)

V_ϕ =

circumferential velocity (m/s)

W =

air inlet width (m)

Y_M =

rate of fluctuating dilatation in compressible turbulence to the overall dissipation rate

r, θ, z =

cylindrical coordinates

ρ =

density (kg/m³)

P_ra =

pressure ratio at the point of extraction (Pa)

t =

time (s)

T =

temperature (K)

v =

velocity (m/s)

η =

efficiency

ν =

kinematic viscosity (m²/s)

μ =

dynamic viscosity (Pa*s)

σ =

prandtl number

$ω$ =

angular velocity (rad/s)

CA =

compressed air

f =

fuel

s =

swirl

_t =

turbine