A Modified Circumferentially Averaged Method for Compressor Performance under Inlet Distortion

Abstract

Inlet distortion detrimentally affects the aerodynamic performance and lessens the stability of compressors, and has received considerable attention. The need to accurately estimate its effect on the performance and stability of compressor as early as possible in a pre-design cycle is emphasized herein. This work presents a modified circumferential average through flow method (CAM) based on parallel compressor (PC) theory for compressor performance and stability analysis under inlet distortion. The PC approach classically considers independent circumferential flow zones evolving through a compressor, with the same outlet static pressure. In the present work, this theory has been modified in order to create a parametric outlet static pressure boundary condition. It enables us to deal with an upstream flow distortion map, and the meridian plane flow field can then be calculated in order to update the corresponding compressor performance. The model is applied in a compressor which has been tested for its performance characteristics under uniform inlet conditions. Utilizing the new model, the stability and performance of the compressor under inlet distortion can be estimated.

1. Introduction

The non-uniform upstream flow of an aero engine and in the intake duct referred to as inlet distortion detrimentally affects the aerodynamic performance and lessens the stability of a compressor. For this reason, the inlet distortion problem has received considerable attention, and the need to accurately estimate its effect on the performance and stability of compressors as early as possible in the pre-design cycle is widely recognized.

For this reason, methods are needed to provide reliable performance prediction during the early design phase of the compressor aerodynamic behavior under distorted inflow conditions in terms of stability margin and efficiency penalty. Some three-dimensional (3D) computational fluid dynamics (CFD) tools have been involved [1,2,3]. However, the 3D CFD simulation is similar to a black box. The simulation outputs are obtained without cognizing the inner mechanism, which makes it difficult to present the analysis of force balance, or to generate quantitative models and empirical formulas. Meanwhile, unsteady disturbances of various scales will appear in the compressor under distortion conditions. The need for fine grid and full annulus computation makes the number of computational costs huge and the computation time too long. When Medic et al. [4] simulated a 20° sector gas turbine engine on the coarse grids containing 14 million cells, they spent nearly 2 weeks relying on 700 processors. In addition, a profound study on other issues, for example, the selection of turbulence model, the setting of boundary conditions, whether it is necessary to introduce and how to introduce the initial disturbance, the reliability of the calculation results, among others, is needed. Thus, the cost and complexity of the simulation remain prohibitive, and CFD tools may not be adapted for preliminary design [5].

This work presents a modified circumferential average through flow method (CAM) based on parallel compressor (PC) theory for compressor performance and stability analysis under inlet distortion. CAM, first proposed by Smith [6], is appealing in the following aspects: choke mass flow can be predicted, certain types of shock can be captured, and the same code can be used for subsonic, transonic and supersonic meridional velocities. Much research has been performed based on CAM in turbomachinery. The PC theory was first introduced by Pearson [7] and has been extensively used to predict the impact of an inlet flow distortion on the stability limit of a fan or a compressor. This approach classically considers independent circumferential flow zones evolving through a compressor, without interacting between each other. In the present work, this theory has been modified in order to create a parametric boundary condition. Instead of relying on a global performance curve, the user inputs are the geometric elements of the compressor and some initial boundary conditions. This enables us to deal with any upstream flow distortion maps, and the meridian plane flow field can then be calculated in order to update the corresponding compressor performance.

2. Methodologies

2.1. Through Flow Analysis Tool

In this research, CAM is used to solve quasi 3D N-S equations with source terms based on the circumferential average dimension reduction method for the meridian plane. For the 3D flow in aviation turbomachinery, the CAM method applied for any parameter is defined as

$$\overline{q}\left(x,r,t\right)=\frac{1}{{\phi }_s-{\phi }_p}{\displaystyle {\int }_{{\phi }_p}^{{\phi }_s}q\left(x,r,\phi ,t\right)}d\phi$$

(1)

where $q$ is an aerodynamic parameter, $\overline{q}$ is the circumferential value, and ${\phi }_s$ and ${\phi }_p$ are angular values for the suction side and pressure side individually. In a vane passage shown in Figure 1, the integral equation starts from the suction side to the pressure side.

A blockage factor due to the circumferential blade thickness takes on the form

$$b=\frac{{\phi }_s-{\phi }_p}{2\pi /N}$$

(2)

where $N$ is the number of blades.

After the definition of CAM, aerodynamic parameters can be decomposed into circumferential average and spatial fluctuation:

$$q\left(x,r,\phi ,t\right)=\overline{q}\left(x,r,t\right)+q^{\prime }\left(x,r,\phi ,t\right)$$

(3)

Based on the CAM and the main operation rule, the N-S equation in the relative frame can be derived as

$$\frac{\partial \overline{U}}{\partial t}+\frac{1}{br}\frac{\partial }{\partial x}\left[br\left(\overline{F}-\overline{F_v}\right)\right]+\frac{1}{br}\frac{\partial }{\partial r}\left[br\left(\overline{G}-\overline{G_v}\right)\right]=\overline{S}+F_B+F_F$$

(4)

where $U$ is the vector of conservative variables, $F$ and $G$ are the convective flux vectors, $F_v$ and $G_v$ are the viscous flux vectors, $S$ is the source term, $F_B$ is the inviscid blade force and $F_F$ is the viscous blade force. More details for CAM may be reviewed in references [8,9].

A time-marching finite volume method is used to solve the equation. The Edwards’ LDFSS [10] scheme is applied to discretize the convective fluxes. The calculation of the viscous fluxes requires the evaluation of the primitive variables, and their first derivatives are at the edge of the middle of the considered cell. The Green–Gauss theorem is also used to estimate the gradient of the variables. An explicit Runge–Kutta method is chosen for the time discretization as it is straightforward to implement. Meanwhile, local time step and implicit residue averaging are used to accelerate the convergence.

The inlet boundary conditions including the inlet total temperature, total pressure and flow direction are given in the calculation of compressible flow. If the Mach number normal to the flow direction is greater than 1.0, then all characteristic waves (u + c, u and u − c) are propagated from the outside into the calculation domain according to the characteristic theory, so the velocity or static pressure should also be given; thus, all flow field variables at the boundary can be determined. Otherwise, if the normal Mach number is less than 1.0, a cluster of characteristic waves (u − c) will propagate outward from the interior of the calculation domain, and a variable must then be interpolated from the interior of the calculation domain. The inlet parameters calculated from the inlet boundary conditions are then assigned to the whole flow field. The wall boundary defines the no slip condition so that the wall velocity is equal to the wall motion velocity.

2.2. Development of the Distortion Method

2.2.1. PC Model

The approach referred to as the PC theory was first introduced by Pearson [7] and has been extensively applied to predict the effect of inlet flow distortion on the stability of a compressor. The idea consists of subdividing the compressor into different sectors along the circumferential direction, considering each of them as independent and fed by a uniform inflow. The original theory assumes a steady behavior and a uniform outlet static pressure distribution, with each PC sector operating at the same rotational speed and different pressure ratios. It is assumed that the flux at any position in PC is equal to the flux at the same local air intake condition under uniform air intake, and varies with the change of local air intake conditions. The air flow satisfies the conservation equations of mass, momentum, energy, and the given inlet and outlet boundary conditions, meaning that no other boundary conditions are attached inside the compressor.

2.2.2. PC with $\mathrm{DC}\left({\mathsf{\theta }}_{\mathrm{crit}}\right)$

The basic PC theory assumes that the stability boundary of the compressor occurs when either PC sector reaches the uniform flow stability limit. Using this approach, the mean operating point at instability is a weighted average of the low flow sector operating at the uniform flow stability boundary. The greatest loss in surge margin occurs with the narrowest spoiled sector inlet distortion, as Longley demonstrated [11].

In the light of experimental observations such as those of Reid [12], the prediction of the large loss for a narrow spoiled sector is inadequate. He presents the reduction in outlet static pressure at surge with the various angles of spoiling during his series of experiments. The boundary condition used for the exit static pressure actually varies with the extent of the spoiled sector. As the angular width of the spoiled sector increases, the exit static pressure of the lower pressure sector drags the overall static pressure of the annulus downwards until it reaches a specific critical width, after which the exit delivery pressure remains roughly constant. This width is referred to as the critical sector angle, ${\theta }_{crit}$ [11].

The concept of a critical sector angle is included within the PC model by specifying that the maximum loss of surge margin occurs when the spoiled sector width is ${\theta }_{crit}$. For distortions narrower than ${\theta }_{crit}$, the loss of surge margin is set to be approximately proportional to $\theta /{\theta }_{crit}$. The implication is that even if a small portion of the compressor annulus operates beyond the usual or natural stability boundary, there is enough of the annulus operating on the stable side to maintain overall stability. This approach is linked to the distortion coefficient (DC) correlation.

The distortion coefficient $\mathrm{DC}\left({\theta }_{crit}\right)$ derived by Rolls Royce is employed to quantify the degree of inlet pressure distortion in a fixed angular region. The coefficient has been proven to be simple and functional [13,14,15], and has been used extensively in the European fighter programs Tornado and Eurofighter [16], and in the GasTurb program [17]. The coefficient is defined as the difference between the mean total pressure for the full $360°$ of the aerodynamic interface plane (AIP) and the average total pressure in the most distorted ${\theta }_{crit}$ sector divided by the average inlet dynamic pressure. It is limited to one-per-revolution, circumferential total pressure non-uniformities:

$$\mathrm{DC}({\theta }_{crit})=\frac{P_{t.av}-P_{t.{\theta }_{crit}}}{\frac{1}{2}\rho V_{av}^2}$$

(5)

The compressor sensitivity is then defined:

$$\mathrm{sensitiviy}=\mathrm{loss} \mathrm{of} \mathrm{surge} \mathrm{margin}/\mathrm{DC}\left({\theta }_{crit}\right)$$

(6)

The critical sector angle is compressor dependent, as the response of compressor to inlet distortion is affected by the geometry characteristics of flow passage and blades. Typically, the angular region for modern compressors is often considered to be $90°$ [18].

2.2.3. Outlet Boundary Correction Model

For each PC sector, the static pressure exit boundary condition is applied. Orme investigated the impact that using a static pressure exit boundary as opposed to a variable-area nozzle had on the observed trends and flow physics by comparing results obtained using both approaches [19]. There was less than 1% difference in the near-stall, near-design, and choke mass flow rates, pressure ratios, and efficiencies. Comparison of total pressure, total temperature, and static pressure contour plots for both approaches showed only subtle differences, which were attributed to the slight differences in operating conditions. He concluded that there was little difference in the overall observed trends when either approach was used. Therefore, simulations using a static pressure exit boundary condition are deemed adequate for comparing distortion transfer and generation in this study. The static pressure is specified along specific blade height (generally on the hub or casing) at the outlet, and then the outlet static pressure distribution along the blade height is solved using a simplified radial balance equation.

$$dp=\frac{\rho V_u^2}{r}dr$$

(7)

For different PC sectors, The PC model assumes that all of them will exit at the same static pressure. Greitzer demonstrated that the assumption can be established only when the last stage stator of the compressor is axially vented and connected with a straight outlet [20]. The investigations by Kurzke [17] and Cousins et al. [21] suggest that assumption is not necessarily true. The outlet may have disparate static pressures in the case wherein cross flow occurs between the sectors when high levels of distortion exists. Additionally, when a compressor is designed closely coupled, there will not be enough room for the circumferential balancing of static pressure, and the downstream field will affect the upstream.

Research has been carried out to take the non-uniformity of the outlet static pressure into account. The obvious solution to the mathematical modeling is to move the PC traditional modeling approach away from the standard method to match the known experimental trends. Ham & Williams [22] developed the coupling number to relate the influence of the downstream component to the size of the non-uniformity at the exit of the upstream component. Cousins [21] approximated the cross flow in the rotor–stator gap using a simple orifice flow analogy developed by Kimzey [23].

Circumferential distortion results in unequal total pressure distribution at the circumferential boundary of PC, which leads to circumferential mixing. Since the distortion scale is much larger than the cascade solidity, it is believed that circumferential mixing mainly occurs in the vane-less region between the rotor and the stator. The circumferential mixing intensity depends on the distortion intensity, sector angle, compressor geometric characteristics and other factors. If the mixing level is low, the flow mixing between the high and low pressure sectors is minor, and the static pressure at the outlet is not equal; the stronger the mixing level is, the more the flow flux between the high and low pressure sectors is mixed, and the outlet static pressures tend to be equal.

Here, the outlet static pressure condition for the PC clean sector ($P_{s.clean}$) and distorted sector ($P_{s.dis}$) is modeled to follow the following equation:

$$P_{s.clean}=\left(1+\mathsf{\beta }·\mathrm{DC}\left(\mathsf{\theta }\right)·L_c\right)P_{s.dis}$$

(8)

$$\mathsf{\beta }=\mathsf{\theta }/{\mathsf{\theta }}_{crit}$$

$$L_c=(L-L_{mix})/L$$

The parameter $\mathsf{\beta }$ represents the extent of the spoiled sector region. $\mathsf{\beta }$ of zero gives a clean annulus case, and as the spoiled sector magnifies, the static pressure at the clean sector will gradually increase. Therefore, the proposed model for determining the boundary condition at extents that are lower than the critical angle is described by $\mathsf{\beta }$, which properly establishes a piecewise linear relationship for the exit static pressure boundary condition as a function of the sector extent. Note that when $\mathsf{\theta }$ is greater than ${\mathsf{\theta }}_{crit}$, the value of $\mathsf{\beta }$ is set to 1.

$\mathrm{DC}\left(\mathsf{\theta }\right)$ is defined in Equation (5). A $\mathrm{DC}\left(\mathsf{\theta }\right)$ of zero gives a clean annulus case (static pressures equal), and as the intensity of distortion increases, the difference between the static pressure at the exit of the clean and spoiled sector increases. When the spoiled sector reaches the critical extent, $\mathrm{DC}\left(\mathsf{\theta }\right)=\mathrm{DC}\left({\mathsf{\theta }}_{crit}\right)$ remains unchanged.

$L_c$ is introduced to account for the influence of the compressor geometric characteristics. L is the axial length of the compressor stage, while $L_{mix}$ is the axial length of the vane-less region between rotor and stator.

3. Application Scheme

3.1. Application Object

The model is applied during the design progress of the axial compressor for 11D turbofan engine. The 11D compressor is comprised of a row of inlet guide vane blades (igv), a row of rotor blades and a row of stator blades. The number of blades accordingly are 3, 15 and 16. The rotor adopts a compound sweep shape, and the stator adopts a curved sweep design. A physical map of the compressor is shown in Figure 2, and the main characteristics are given in

Table 1. 11D axial compressor characteristics.

Parameter	Value
Design rotational speed (rpm)	22,000
Mass flow rate (kg/s)	21
Pressure ratio	1.6
Efficiency	0.865

.

3.2. Comparative Data

A major obstacle encountered during the application process is the lack of experimental results against which the model results are compared. In the literature that contains experimental investigation on distortion, there is inadequate information provided, so the case could not be replicated. Therefore, the test data obtained from the open literature are mainly used to verify the trends. The actual experiments on the 11D turbofan engine under uniform inlet boundary conditions have been performed along with the single passage 3D CFD simulation under uniform inlet flow for comparison with the improved CAM. For the inlet distortion, the steady full annulus 3D CFD simulation which has been proved reliable in the comparison with experimental result is conducted for validation of the proposed method. In addition, further distortion experiments on the 11D turbofan engine may be performed to certify the results of the improved CAM in future research.

3.2.1. Experiment on the 11D Turbofan Engine under Uniform Inlet Boundary Condition

The results of this experiment are used for comparison with the 3D CFD simulation and CAM to prove the reliability of the two numerical methods. The experimental method is briefly introduced here.

Figure 3 and Figure 4 display the test bench and the general schematic of the 11D turbofan aero engine. The meanings of location index in Figure 4 are as follows: 0—measurement section of inlet total temperature, total pressure and static pressure, 1—measurement section of total pressure, total temperature and static pressure in front of compressor rotor, 2—measurement section of total pressure, total temperature and static pressure behind the first stator, 3—measurement section of total pressure, total temperature and static pressure after centrifugal compressor, 4—total pressure and total temperature measurement section in front of turbine guide vane, 5—measurement section of total pressure and airflow direction after turbine rotor, 6—total pressure and temperature measurement section of nozzle, and 16-measurement section of total pressure, total temperature and static pressure at the outlet of the bypass. The measuring content and distribution of measuring points are shown in

Table 2. Measuring content and distribution of measuring points (number of probes × number of span-wise measuring points).

Probe Type	Total Pressure	Total Temperature	Outer Wall Static Pressure	Internal Wall Static Pressure
Measuring Section 1	1 × 5	1 × 5	1	1(0)
Measuring Section 2	2 × 6	2 × 6	4	1(0)
Measuring Section 6	1 × 5	1 × 5	0	0

and Figure 5, taking measurement Section 1, 2, and 6 for example.

In addition to the total temperature and total pressure parameters obtained from each measurement section, the measurement method of the engine performance parameters is as follows: the thrust is measured by the thrust sensor installed on the test bench. The rotational speed is obtained by the speed sensor. The fuel flow is measured by the turbine flowmeter connected to the fuel supply system. The discharge temperature is measured by the thermocouple at the nozzle. The acceleration sensor for measuring vibration signal is arranged at the main mounting joint for engine vibration. The air flow is calculated according to the inlet total temperature, total pressure and wall static pressure. The pressure ratio is calculated by averaging the total pressure values at the span-wise measuring points of the inlet and outlet sections.

The average of the data at the same measuring point and the circumferential average of the parameters are both obtained by the arithmetical average method, so that the average span-wise distribution of the parameters (total temperature, total pressure) and the average section parameters (static pressure) can be obtained. The acquisition of section average parameters requires an area average of span-wise distribution parameters. In this study, the parameters between adjacent span-wise positions are assumed to be linearly distributed. The parameters are assumed to be uniformly distributed in the boundary layer, and equal to the nearest measuring point to the boundary. The area average method of parameters is as follows, taking the total pressure as an example:

$$\begin{array}{ll}\overline{P_i^{*}}& =\frac{{\displaystyle \int P^{*}}dA}{A}\\ & =\frac{{\displaystyle \sum _{j=0}^J{\displaystyle \int \left(\frac{r-r_{j+1}}{r_j-r_{j+1}}·P_{i,j}^{*}+\frac{r-r_j}{r_{j+1}-r_j}·P_{i,j+1}^{*}\right)·2\pi rdr}}}{A}\\ & =\frac{{\displaystyle \sum _{j=0}^J\frac{\pi }{3}(P_{i,j}^{*}+P_{i,j+1}^{*}+\frac{P_{i,j}^{*}{r}_j+P_{i,j+1}^{*}{r}_{j+1}}{r_j+r_{j+1}})(r_{j+1}^2-r_j^2)}}{\pi (r_{J+1}^2-r_0^2)}\end{array}$$

(9)

where $\overline{P_{i,0}^{*}}=\overline{P_{i,1}^{*}}$, $\overline{P_{i,J+1}^{*}}=\overline{P_{i,J}^{*}}$, i represents the serial number of the measuring section, and j represents the serial number of the span-wise position of the measuring point.

Subsequently, the mass flow rate, pressure ratio, efficiency and other parameters can be calculated according to the temperature and pressure acquired at each section. Taking the calculation of compressor inlet mass flow rate as an example:

$$m_f=K_{G}K{A}_0\frac{P_0^{*}}{\sqrt{T_0^{*}}}q\left(M{a}_0\right)$$

(10)

where $K_G$ is the flow coefficient of the flow pipe, considering the boundary layer effect (this paper takes the empirical value of 0.994 for correction).

3.2.2. 3D CFD Simulation on the 11D Turbofan Compressor

Two cases of 3D CFD simulation have been performed on the 11D compressor, one single passage case for the clean inlet flow condition and the other full annulus case for the distorted inlet condition. The clean inlet condition case is compared with the experimental results to verify the reliability of the 3D CFD simulation and then, based on the verification, the full annulus case with inlet distortion is used to verify the improved CAM in this study. The 3D CFD simulation method is briefly introduced here.

For both cases, the multi-block structured grid technology, about 900,000 nodes for the single passage case and 9,300,000 nodes for the full annulus case, is adopted to discretize the computational domain, as shown in Figure 6. The tip clearance of the rotor is set to 0.5mm from the leading edge to the trailing edge. The O4H topology is applied to the mesh generation of the igv, and the HOH topology is adopted for the rotor and the stator. The y+ distribution is a significant criterion for the grid, and is observed to be less than 5 after the calculation result is obtained.

The steady-flow fields corresponding to a series of valve positions along the throttling process are computed through solving Reynolds-averaged Navier–Stokes equations, closed by a Spalart–Allmaras (S-A) turbulence model. The S-A model is known to predict premature stall when applied to a compressor, and has been proven reliable in turbomachinery applications due to its simplicity, robustness, and efficiency [24]. The rotor-stator interfaces are set to conservative coupling by pitch-wise row. A fixed inlet boundary condition is set with total temperature (288.15 K), total pressure (101,325 Pa for the clean inlet condition case and a distortion map for the distorted case), and velocity direction (normal to the inlet section). The throttling process is simulated by gradually increasing the outlet static pressure. The radial pressure distribution at the outlet boundary is governed by radial equilibrium condition. As for the body surfaces, no slip velocity and adiabatic boundary conditions are applied.

The grid independence is verified before the study. For the single passage case, three types of mesh with different numbers of grid nodes are generated. The number of nodes for the three grid types are 720 k, 904 k, and 1050 k, respectively. The topology of the mesh and the boundary conditions are set as the same. The performance characteristics of the compressor based on these three grids, including the pressure ratio versus the mass flow and the adiabatic efficiency versus the mass flow, are obtained and plotted in Figure 7. The curves of the case with 904 k grid nodes coincide with those of the case with 1050 k grid nodes. In conclusion, when the number of grid nodes exceeds 904 k, the result is considered independent of the number of grid nodes. In consideration of the calculation accuracy and the requirement of saving time, the single passage case with 904 k grid nodes is introduced into this study.

Meanwhile, three types of mesh with different numbers of grid nodes are generated for the full annulus case, and the number of nodes are 8.54 million, 9.33 million and 9.61 million, respectively, with the same mesh topology and boundary condition. The pressure ratio versus the mass flow and the adiabatic efficiency versus the mass flow are shown in Figure 8. It is observed that when the number of grid nodes exceeds 903 million, the result is considered independent of the number of grid nodes. In consideration of the calculation accuracy and the requirement of saving time, the full annulus case with 903 million grid nodes is employed in this study.

3.2.3. Application of the Improved CAM on 11D Turbofan Compressor

Several cases of CAM have been performed on the 11D compressor. One clean inlet condition case gained from the origin CAM is compared with the experimental and 3D CFD results to verify the reliability of the method. Additionally, then, based on the verification, two cases with inlet distortion (obtained by the origin CAM and the improved CAM, respectively) are used to analyze the performance and stability of 11D compressor with inlet distortion. The calculation process is briefly introduced here.

For all cases, the meridian computational grid topology of 36 × 27 (Igv) + 36 × 27 (Rotor) + 25 × 27 (Stator) + 100 × 27 (Outlet) nodes is adopted to discretize the computational domain, as shown in Figure 9.

The steady flow fields corresponding to a series of valve positions along the throttling process are computed through solving an N-S equation modeled by CAM. A fixed inlet boundary condition is set with total temperature (288.15 K), total pressure (101,325 Pa for the clean inlet PC sector and 960,000 Pa for the distorted PC sector), and velocity direction (normal to the inlet section). The throttling process is simulated by gradually increasing the outlet static pressure. The radial pressure distribution at the outlet boundary is governed by radial equilibrium condition.

To check the grid independence in the CAM simulation of the 11D compressor, three grids are made with grid numbers of 3795, 5319 and 7161, respectively. The performance characteristics of the compressor based on these three grids are obtained and shown in Figure 10, including the pressure ratio versus the mass flow and the adiabatic efficiency versus the mass flow. The performance line of the case with 5319 grid nodes coincide with that of the case with 7161 nodes. Hence, in conclusion, the results are considered independent with the grid number when the number of grid nodes exceeds 5319. In consideration of the calculation accuracy and the requirement of saving time, the numerical method with 5319 grid nodes is involved in this study.

3.2.4. Comparison of Various Methods under Uniform Inlet Condition

The comparison among the actual experiment, 3D CFD simulation and the origin CAM computation on the total pressure ratio and efficiency of the 11D turbofan engine at 100% speed under uniform intake airflow is shown in Figure 11 and Figure 12. The pressure ratios obtained by both simulation approaches are lower than test results. Specifically, at the operating point, the relative error of mass flow between the 3D CFD and test is −1.14%, the relative error of the total pressure ratio is −1.80%, and the relative error of the efficiency is −4.69%. Additionally, for CAM, the values are −1.11%, −0.72% and −4.75%. Overall, both computational characteristics match the experimental results well, while the full annulus 3D CFD simulation has the ability to calculate the performance and stability of compressor under inlet distortion. Therefore, the 3D CFD simulation may be used to verify the accuracy of the improved CAM on the performance and stability calculation of compressor under inlet distortion in this study. Other test cases from open publications are focused on verifying major trends.

3.3. Distortion Descriptor

There are various methods for determining the impact of the inlet distortion on compressors in different references. According to the previous analysis, the $\mathrm{DC}\left(90\right)$ is employed to quantify the inlet total pressure distortion. The total pressure distributions at the inlet aerodynamic interface plane (AIP) of all calculation cases in this study are given as shown in Figure 13, in order to simplify the study process. The inlet total pressure of the clean sector is 101,325 Pa, and the distorted sector is 96,000 Pa; the average total pressure is 99,994 Pa. The specific value of $\mathrm{DC}\left(90\right)$ can be obtained depending on the dynamic pressure.

Figure 14 acquired from the full annulus 3D CFD simulation on the 11D compressor with inlet distortion, shows the correlation of $\mathrm{DC}\left(90\right)$ versus the mass flow under 100% rotational speed under the condition that the inlet total pressure distribution is determined. When the mass flow decreases, the axial velocity of the inlet AIP decreases, the dynamic pressure decreases, and according to the equation of $\mathrm{DC}\left(90\right)$, $\mathrm{DC}\left(90\right)$ increases. The figure shows that the $\mathrm{DC}\left(90\right)$ parameter is not a fixed value, but varies with the local condition. It reflects the intensity of total pressure distortion on one hand and the intensity of the dynamic pressure distortion as well.

4. Result and Discussion

4.1. Performance Characteristics

The characteristic curves of the 11D compressor at 100% revolution speed under inlet distortion are calculated by full annulus 3D CFD simulation, the origin CAM and the improved CAM, respectively.

The curve of pressure ratio versus mass flow under the influence of total pressure distortion at 100% speed is shown in Figure 15. The blue curve is from single passage 3D CFD simulation with clean inlet flow for comparison, while the yellow line is from steady full annulus 3D CFD simulation but with distortion. The grey curve shows the result of the origin CAM with constant exit static pressure, and the orange curve shows the result of the improved CAM with corrective exit static pressure obtained from Equation (8). The three performance curves obtained under the inlet distortion condition have basically the same trends. Compared with the uniform inlet characteristic curve, the total pressure ratio under the total pressure distortion at the near stall point decreases and the flow rate increases. The circumferential total pressure distortion makes the characteristic curve of the compressor move downward. With the decrease in flow rate, the ability of the compressor to overcome the distortion becomes weaker, and the performance deteriorates more seriously. Among them, the total pressure ratio loss calculated by steady full annulus 3D CFD simulation is the largest. The stability boundary is under-predicted by the CFD approach due to the inability of mixing planes to deal with asymmetric flows [25]. Meanwhile, stiff outlet boundary condition or the S-A turbulence model [26] may be the reason for the divergence of numerical simulation of high-speed compressor with low flow rate.

The origin CAM adopts the PC theory, which assumes that the same outlet static pressure is given for both clean sector and distorted sector, and the stability boundary of the compressor occurs when distorted sector reaches the uniform flow stability limit. The stability boundary obtained by the method is more abundant than that of steady full annulus 3D CFD simulation, which is more consistent with the characteristics of compressor.

Furthermore, the improved CAM assumes that different sectors have disparate static pressures depending on the distortion intensity, distortion range and specific compressor characteristics. By introducing outlet static pressure correction formula, the stability boundary gains a more realistic range.

The curve of efficiency versus mass flow under the influence of total pressure distortion at 100% speed is similar to the result of pressure ratio, as shown in Figure 16. The results of various calculation models on the operating line are relatively consistent. At the stability boundary, the improved CAM has a smaller flow rate at the surge point, and obtains a stability boundary that is more consistent with the characteristics of the compressor than the origin CAM and the 3D CFD method.

The compressor stability margin defined at constant rotor speed has advantages for the engine manufacturer; most compressor design procedures and testing are carried out at constant rotor speed. Additionally, the stability margin at limiting rotor speed can be defined without extrapolation [27].

$$\mathrm{SM}=\frac{P{R}_{sta}/W{A}_{sta}-P{R}_{ope}/W{A}_{ope}}{P{R}_{ope}/W{A}_{ope}}\times 100\%$$

(11)

Loss in stability pressure ratio due to inlet total pressure distortion ($\Delta \mathrm{PRS}$) is the loss in stability pressure ratio due to inlet distortion normalized by the undistorted stability pressure ratio.

$$\Delta \mathrm{PRS}=\frac{P{R}_{clean}-P{R}_{dis}}{P{R}_{clean}}\times 100\%$$

(12)

The SM with a clean inlet boundary condition is 11.9%. At the stability boundary under inlet distortion, the $\mathrm{DC}90$ is 0.289, SM is 6.2%, $\mathrm{and} \Delta \mathrm{PRS}$ is 2.7%. When the operating point is located at the minimum flow point on the distorted performance line, the circumferential total pressure distortion consumes the compressor stability margin and has a significant impact on the compressor stability.

The stability boundaries of the 11D compressor under the uniform inflow and inlet distortion are obtained using the improved CAM, as shown in Figure 17. The blue solid line is the calculation result of 11D compressor at 90%, 95% and 100% revolution speed under uniform inflow. The gray short dotted line is then obtained, showing the stability boundary with uniform inflow. The orange solid line is the calculation result at 90%, 95% and 100% revolution speed under inlet distortion, and the yellow long dotted line indicates the stability boundary with inlet distortion. According to the calculation result of the improved CAM, the compressor performance is affected by inlet distortion at different speeds. At the same mass flow rate, the pressure ratio decreases significantly, and the stability boundary shifts to the lower right.

4.2. Span-Wise Distribution of Performance Parameters

The span-wise distribution of rotor pressure ratio and adiabatic efficiency, the stator total pressure recovery coefficient of the 11D compressor at operating point with inlet distortion obtained by full annulus 3D CFD simulation, and the improved CAM are compared in Figure 18 respectively. It reveals that the trend of adiabatic efficiency obtained from the improved CAM is close to the full annulus 3D CFD simulation result, and the trend of rotor pressure ratio and stator total pressure recovery coefficient remains nearly consistent. The rise in pressure in the middle area of the rotor blade is larger, and the root and tip are smaller, which conforms to the design intention. The total pressure recovery coefficient in the tip area of the stator is relatively lower, is reasonably distributed and meets the design expectation. The result of the improved CAM is reliable.

The comparison of the span-wise parameter distribution of the rotor and stator between the clean sector and the distorted sector at the stability boundary obtained by the improved CAM is shown in Figure 19. Affected by the total pressure distortion, the absolute values of inlet and outlet total pressure of the rotor in the distorted sector are lower than that of the clean sector, while the ratio is greater than that of the clean sector. The distorted sector of the rotor is operating near the stability boundary condition, from which a flow separation is more easily derived, and its efficiency is thus lower than that of the clean sector. The stator is affected by the total pressure distortion from upstream, and the total pressure recovery coefficient in the distorted sector is thus lower than that in the clean sector.

4.3. Axial Transmission of Distortion

In order to study the propagation mechanism of inlet distortion along the axial direction of the compressor, the circumferential distortion intensity (CDI) parameter is introduced here. CDI is the parameter recommended by the Society of Automotive Engineers (SAE) to evaluate the total pressure distortion intensity. For detailed introduction, please refer to reference [27]. Since this study focuses on the axial transmission of distortion, which has little relationship with the radial distribution, the equation is simplified here as follows:

$$\mathrm{CDI}=\frac{P_{t.av}-P_{t.dis}}{P_{t.av}}$$

(13)

By calculating the operating point of 11D compressor at 100% speed with the improved CAM, the transmission characteristics of total pressure distortion along the axial direction is gained, as shown in Figure 20. The total pressure distortion attenuates rapidly when passing through the leading edge of the rotor, and the attenuation capacity of total pressure distortion is different for different blade heights. The total pressure distortion attenuation at the blade hub is smaller than that of the blade middle and tip; Though the distortion is enhanced to some extent when passing through the rotor region, the overall distortion is weakened compared to the initial distortion. It shows that the rotor has a certain attenuation effect on total pressure distortion. There is a certain loss of total pressure when passing through the stator, so rather than inhibiting the total pressure distortion, the stator increases the total pressure distortion intensity.

4.4. Meridional Flow Field Distribution

The meridian flow field distribution of total pressure, total temperature, static pressure, axial velocity, absolute Mach number and absolute air flow angle of distorted sector and clean sector at the stability boundary of 11D compressor simulated with the improved CAM are shown from Figure 21, Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26 to illustrate the transfer process of distortion in compressor.

When the airflow passes through the rotor stage, the rotor does work to increase the total pressure of the airflow, and the total pressure distortion is suppressed after passing through the rotor. When the air flows through the stator, there is a certain loss of total pressure which increases the distortion intensity rather than weakening it.

Although the inlet boundary condition of the compressor is total pressure distortion and the total inlet temperature is uniformly distributed, there are obvious total temperature distribution differences between the distorted and clean sectors in the hub region. The distorted sector is affected by the total pressure distortion and the loss is large. From the previous analysis, it can be seen that the loss in the tip area is greater, thus generating obvious total temperature distortion in this area.

At the stability boundary, the flow angle at the rotor tip in the distorted sector is greater, which increases the angle of incidence, Mach number, flow loss, total temperature and distortion intensity, and the pressurization capacity at the rotor tip is weakened. It is most likely that large flow separation will occur first at this working condition, making the compressor enter an unstable operating state.

These phenomena are consistent with the existing cognition, which shows that the improved CAM accurately simulates the effect of total pressure distortion on the performance of compressor, and correctly reflects the main flow field structure under total pressure distortion.

5. Conclusions

An improved CAM based on PC theory for compressor performance analysis under inlet distortion is proposed, and the method is applied to analyze the aerodynamic characteristics and stability of 11D compressor under circumferential total pressure distortion. This model can effectively reveal the aerodynamic characteristics of the compressor under inlet distortion, as well as the characteristics of the radial and axial flow fields. The main conclusions are as follows.

The precondition of identical outlet static pressure for each PC sector in the PC model is not consistent with the physical phenomenon. An outlet static pressure correction model is applied in CAM to enhance the stability boundary prediction accuracy for compressor under inlet distortion.

Using the improved CAM and based on PC theory, the stability parameters and aerodynamic performance parameters, including the span-wise distribution curve and meridional flow field distribution of the 11D compressor with inlet distortion, are obtained and analyzed. The circumferential total pressure distortion seriously aggravates the performance and stability of the compressor. Compared with the uniform intake results, the 90-degree circumferential total pressure distortion with $\mathrm{DC}\left(90\right)$ varies from 0.263 to 0.289, or the outlet static pressure reduces the total pressure ratio at the near stall point by 0.8%, increases the flow rate at the near stall point by 2.0%, and decreases the stability margin by 5.2%. The total pressure distortion intensity is obviously weakened after passing through the rotor, but enhanced after passing through the stator to some extent. The high loading at the rotor tip caused by the high angle of incidence greatly dilutes the ability of the rotor tip to overcome the flow separation, and increases the flow loss. In this area, it is easy for flow separation to occur first, which reduces the aerodynamic stability of the compressor. Meanwhile, the inlet total pressure distortion increases the rotor work in the distortion region, which causes high total temperature distortion at the rotor outlet. The attenuation effect of the compressor on distortion is significant; on the other hand, it is sensitive to distortion and prone to aerodynamic instability. Special attention should be paid during the design process.

The method proposed in this paper for inlet total pressure distortion reflects the mechanism of the inlet distortion affecting on the performance and stability of the compressor, and reveals the transmission characteristics of the distortion along the compressor interior. The simulation results of the method are close to the full annulus 3D CFD simulation results and consistent with the physical phenomena, and prove the accuracy of the method. The method has the ability to simulate the response of a general axial flow compressor to steady inlet distortion.