Governing equations

The governing equations for the body force approach are those for mass (2), momentum (3) and energy conversation (4), (Liepmann, 2013):

In the momentum equation the body force f gets added within the term SM=ρf. The same applies for the energy equation with SE=ρf⋅U where the blades rotation and thus its work input is represented by its blade velocity U=ω⋅r. In case of a stator there is no rotation, which means there is no work input SE=U=0.

Numerical setup

Due to the nature of the body force model, there is no circumferential variation in the model equations. The blade row gets modeled in an axisymmetric way. A comparison of the numerical setups of a bladed simulation versus a body force simulation is pictured in Figure 3.

Figure 3.

Comparison of numerical setups, Blade domain (left) and BFM domain (right).

Figure 3 illustrates a key advantage of the body force approach over traditional bladed domain simulations. In the bladed domain, it is necessary to have a high spatial discretization of boundary layers. On the other hand, the BFM domain can be spatially resolved much more coarsely, since there is no need to resolve the blade boundary layer. To create the mesh for the BFM domain, a 2D mesh is created in the meridional plane and then rotated in the circumferential direction, thus adding cells in the circumferential direction. This requires only information about the endwall geometry, as well as the location of the leading and trailing edge.

Unlike the axisymmetric nature of the BFM setup, the model can react locally to non-axisymmetric inflows. Therefore, the model is able to accurately simulate the effects due to inflow disturbances.

State of the art

The calculation of the two body force components f→ can be achieved in several ways. Originally developed by Gong (1999), the formulation has undergone modifications over time. Recently, Hall et al. (2017) proposed a new formulation of the normal component, which only depends on local flow variables. This eliminates the need for empirical calibration, which was necessary in previous studies such as (Gong, 1999; Peters, 2014).

Hall’s model uses the blade camber surface geometry as a priori input and can be expressed as follows:

The local inviscid body force is determined by equating the local deviation angle δ, local camber surface normal vector nθ, relative flow velocity W and blade surface area, distributed over the local blade pitch s.

Although this model was capable of predicting the inviscid redistribution effect of a BLI propulsor, it could not predict efficiency, as Hall neglected the fp component. Additionally, the model did not account for blockage effects, resulting in poorer performance for off-design conditions. However, Hall et al. (2017) and Akaydin and Pandya (2017) applied this model to several fan stages with inflow distortion for on-design conditions, demonstrating the model’s capability to affordably and accurately represent fan stage aerodynamics with inflow distortion.

Further improvements were made by Thollet (2017) by

Loss model

The main limitation of Hall’s model was its inability to capture losses. Thollet uses a simple formulation which relies on the local friction coefficient Cf, which is derived from an empirical turbulent flat plate correlation and uses the local Reynolds number Rex.

This leads to the loss fomulation:

The off-design term in Equation (8) is determined using the distribution of the deviation angle δηmax(x,r) at peak efficiency. To estimate the performance at operating points away from peak efficiency, the local deviation distribution is compared to δηmax(x,r), controlling the slope of the compressor speedline towards stall and surge.

Blockage effects

In order to account for metal blockage effects and its influence on the choking mass flow, based on the reduction in effective flow area due to the presence of physical blades this has to be modeled within the body force approach. Therefore, a blockage factor b is introduced. The blockage factor b is a dimensionless factor that equates the camber to camber pitch distance s of two neighboring blades to the geometrical distance between pressure and suction side of these two blades. This parameter accounts for the reduction of effective area within a blade passage, as pictured in Figure 4.

Figure 4.

Definition of blade metal blockage parameter b.

It is calculated by

The blockage factor is included within the governing Equations (2)–(4):

Compressibility correction

The original model by Hall was tested on incompressible test cases. Thollet has proven, that the following correction yields a valuable impact for transonic configurations. The baseline model of Hall is based on the lift coefficient of a thin airfoil at low Mach numbers, but the lift coefficient of an airfoil increases at high subsonic Mach numbers and has a different expression for supersonic flows. Therefore, a compressibility correction is added

which is an derivation of the Prandtl-Glauert correction for subsonic mach numbers and of the Ackeret formula for supersonic mach numbers. In order to avoid the singularity near Ma=1, a blending function is used (Thollet, 2017).

Resulting equations

These modifications lead to the following model formulation for the inviscid and viscous body forces (Thollet, 2017):

The necessary input for the model is the distribution of the metal angle κ(x,r) and the blockage factor b(x,r). the deviation angle distribution δηmax(x,r) can either come from a first BFM simulation at peak efficiency or from a bladed simulation. This models capabilities will be discussed in the following and compared to a bladed model.

Implementation in solver

The results presented in this paper were obtained through the use of ANSYS CFX 21R2. ANSYS CFX uses a coupled solver, which solves the hydrodynamic equations as a single system with a fully implicit discretization of the equations (ANSYS Inc, 2021). In order to use the body force approach in CFX, it is necessary to divide the domain into several subdomains. The subdomains representing the rotor and stator have additional sources corresponding to fn and fp. These forces are added as General Momentum Sources to the momentum equation and as an additional Energy Source to the energy equation with SE=ωr(fn+fp). The effects of metal blockage are accounted for, by making use of porous domains in CFX. In a porous domain, there is a virtual solid, that reduces the flow volume by limiting the volume V′ available to flow in an infinitesimal control cell surrounding the point, and the physical volume V of the cell.

Local blockage can be modeled, by defining a local volume porosity γ(x,r) equal to the blockage factor b(x,r).

Blockage model

The blockage model was tested using a generic NACA0016 geometry (Eastman et al., 1933) as a stator. Since the geometry is a zero lift profile, there are no resulting inviscid body forces fn since the blade is not cambered, thus resulting in κ=0. Additionally, to differentiate the effect of the metal blockage, viscous forces were neglected. For turbulence closure, the k–ω-model by Wilcox (2008) is employed. The test case setup is shown in Figure 5(a). Figure 5(b) shows the associated distribution of the blockage factor b(x,r).

Figure 5.

Stator test case. (a) Numerical setup of NACA0016 stator test case. (b) Blockage Factor distribution of NACA0016.

The distribution in Figure 5(b) shows the geometric relationships for the blockage factor. Towards the leading and trailing edge the values of blockage factor approach b=1. Close to the hub the values are the lowest because of the smallest circumferential distance between two neighboring blades while the distance towards the shroud increases due to the larger radius of the circle segment. A comparison of the body force approach against the bladed simulation is made in Figures 6 and 7.

Figure 6.

Contour plots of the Mach number at midspan for the NACA0016 stator test case. Subsonic conditions (left) and transsonic conditions (right). (a) Bladed Simulation with subsonic conditions. (b) Bladed Simulation with transsonic conditions. (c) BFM with subsonic conditions. (d) BFM with transsonic conditions.

Figure 7.

Circumferentially averaged Mach number at midspan from inlet to outlet for the NACA0016 stator test case. Subsonic conditions (left) and transsonic conditions (right). (a) BFM vs. Bladed Simulation with subsonic conditions. (b) BFM vs. Bladed Simulation with transsonic conditions.

Figures 6(a) and 6(c) illustrate the Mach number distribution for subsonic flow conditions. The axisymmetric nature of the body force model is evident in Figure 6(c), where no gradients in the Y direction are present. Both models accurately capture the flow acceleration in accordance with the Bernoulli equation, as demonstrated in Figure 7(a), where the circumferential averaged Mach number at midspan from inlet to outlet is compared. The BFM approach is capable of reproducing the magnitude, location, and slope of the Mach number peak, except for the onset of acceleration at the leading edge, which is not perfectly captured due to the missing stagnation point and its effect on the upstream flow in the BFM approach.

For transonic flow conditions, Figures 6(b), 6(d), and 7(b) depict the Mach number distribution. Similarly, the BFM approach accurately predicts the flow behavior, including the position of the shock and the faster flow downstream of the stator. However, the strength of the shock is underestimated, and the onset of acceleration at the leading edge of the stator is slightly underestimated as well.

These results demonstrate that the BFM approach, developed from Equations (10)–(12), can effectively capture metal blockage effects for the inviscid case. However, it should be noted that the validation of metal blockage has only been conducted for the inviscid case. For boundary layers due to viscous walls, the effective area between two neighboring blades decreases even more, which increases blockage due to aerodynamic effects. This aspect has not been addressed in this study but is left for future investigations. In addition, a relatively straightforward calculation was utilized to determine the blockage factor b. In previous studies, Baralon (2000) proposed an alternative approach to compute the blockage factor b. Unlike the approach adopted here, Baralon calculates the blockage factor perpendicular to the camber surface. Notably, Baralon was able to achieve encouraging results with the NASA 67 rotor. This approach is planned to be investigated further in future work.

Transsonic fan stage

In the following, the model is tested using a transonic fan stage. For this test case the CA3ViAR (Composite fan Aerodynamic, Aeroelastic, and Aeroacoustic Validation Rig) fan stage is used (Eggers et al., 2021) which features a composite low-transonic fan. Its fan stage characteristics are shown in Table 1.

To evaluate the accuracy of the model, a bladed simulation is conducted in conjunction with the BFM simulation. The numerical configurations for both simulations are displayed in Figure 8(b), with identical boundary conditions applied. The inlet is subjected to total conditions, while the outlet is maintained at a static pressure with radial equilibrium. With the exception of the horizontal hub in front of the spinner, all walls are assigned as viscous walls. For turbulence closure, the k−ω-model by Wilcox (2008) is employed. The bladed setup implements a mixing plane between the rotor and stator domains, which is unnecessary for the BFM setup. A comparison of the mesh statistics for both setups is presented in Table 2. The BFM setup has approximately one-fourth of the cell count of the single-passage bladed setup, and this can be further reduced since the BFM mesh has not been optimized yet. The primary objective of this study is to evaluate the precision of the BFM simulation in comparison to the bladed simulation, and mesh optimization will be considered in future work.

Figure 8.

Numerical setups for CA3ViAR fan stage. (a) Bladed Simulation. (b) BFM.

Figure 9 compares the results of bladed simulations and the body force approach for the Cruise and T/O operating points listed in Table 1. Speedlines were calculated for each operating point, and the total pressure rise over the fan πt,fan and the polytropic efficiency ηpoly were plotted against the mass flow rate. The black line represents the bladed simulation data, the orange line represents the data obtained using the BFM approach, and the blue line represents the data obtained using the BFM approach without the off-design term in Equation (8).

Figure 9.

Compressor Map for the CA3ViAR fan stage test case, pressure ratio πt,Fan (left) and polytropic efficiency ηpoly (right).

The results demonstrate that the BFM approach accurately predicts the choke mass flow for both operating points. However, at lower mass flow rates, the BFM approach slightly overestimates the total pressure rise. The impact of the off-design term from Equation (8) can be observed by comparing the blue line to the orange line, which shows that the design term causes the slope of the speedline to decrease from the point of maximum efficiency towards lower mass flows. Without this term, the efficiency and work of the fan would be overpredicted by the BFM approach. However, even with the off-design term, Figure 9 shows that the efficiency is overpredicted by approximately 3% by the BFM approach. This overprediction decreases towards lower mass flow rates, which is consistent with the findings in (Thollet, 2017). The BFM approach currently cannot capture the effects of tip clearance present in the physical blade row. The model assumes no tip gap exists and adds work in this region, increasing the pressure ratio and polytropic efficiency.

Figure 10 presents the local result quality of the BFM approach, where the spanwise distributions of the work coefficient and isentropic efficiency downstream of the rotor are shown in Figure 10. These figures demonstrate that the BFM approach provides satisfactory results for the maximum efficiency operating point at the T/O operating condition. However, discrepancies in the model are evident in predicting the work input of the rotor accurately. The BFM overestimates the work input close to the hub and underestimates it in the direction towards the shroud, which is consistent with the findings in (Thollet, 2017). The lack of consideration of the tip gap does not justify this deviation. To further investigate this, a bladed simulation without a physical tip gap was performed, and the results are shown in the orange line. As expected, the absence of the peak gap increases the work input close to the shroud compared to the bladed simulation with a tip gap. However, this does not explain the spanwise distribution of the BFM approach. The isentropic efficiency in Figure 10 also shows similar behavior, where the BFM approach overestimates the isentropic efficiency near the hub and shroud but agrees precisely with the results in the core region of the flow.

Figure 10.

Spanwise profiles at peak efficiency for T/O conditions downstream of the rotor, Work coefficient (left) and isentropic efficiency (right).

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