# Comparing Blade-Element Momentum Modeling to 3-D CFD

### Many small unmanned aerial vehicles (SUAVs) are driven by small-scale fixed-blade propellers, and the flow produced by the propeller can have a significant impact on the aerodynamics of the SUAV itself.

Small unmanned aerial vehicles (SUAVs) are becoming increasingly popular for surveillance and numerous other applications. These SUAVs come in various sizes, and the smallest are referred to as micro aerial vehicles (MAVs). For purposes here, SUAV will be used to refer to all UAVs that are portable by a man.

SUAVs commonly use small-scale fixed-blade propellers for propulsion. Fixed-blade propellers means the blade is rigidly fixed to the hub so that the blade pitch cannot be changed for various flight conditions. Propellers mounted in a tractor configuration often have significant effects on SUAV aerodynamics. Therefore, to perform Computational Fluid Dynamics (CFD) simulations of a SUAV-propeller system, the SUAV and the propeller must often be simulated in a coupled fashion as the SUAV-propeller interaction is strong. In the design and analysis of a SUAV, hundreds of SUAV-propeller coupled CFD simulations are needed. Performing high-fidelity, time-dependent 3-D Reynolds-averaged Navier-Stokes (RANS) CFD simulations in which the propeller is rotated relative to the aircraft is very expensive computationally. For compactness, this method will be referred to here as the high-fidelity blade model (HFBM). HFBM is an unsteady problem, therefore steady-state convergence acceleration techniques cannot be used.In addition, the fine grid needed to resolve the detailed flow around the propeller blades makes the overall grid size extremely large. HFBM is the most accurate and high-resolution method of propeller modeling as all the 3-D, compressibility, rotational, transitional, and turbulence effects are modeled. However, the high computational cost of HFBM makes it infeasible when numerous simulations are needed, as is the case for many SUAV-propeller problems.

For computational efficiency, steady-state models approximate the time-average flow produced by a propeller. These models embed momentum source terms into the propeller region of a mesh to induce thrust and swirl into the flow field. Many of these momentum source models are based on blade-element momentum theory (BEMT). BEMT determines the thrust and swirl from 2-D airfoil data. However, flow around small-scale propellers can be very complex and highly 3-D in nature, making it difficult for BEMT to accurately predict the propeller performance in many instances.

For this study, researchers from Mississippi State University compared HFBM simulations to a BEMT model for two small-scale propellers to determine the validity of using BEMT to model small-scale propellers in a wide range of flight conditions.

#### High-Fidelity Blade Modeling

HFBM simulations were conducted with an in-house code at MSU called CHEM. CHEM is a second-order accurate, cell-centered finite volume CFD code and has been validated and applied to a wide range of problems. All HFBM simulations were compressible, viscous, and assumed to be turbulent using Menter's shear stress transport (SST) turbulence model. While the Reynold’s number was low (<150,000), the SST turbulence model was used to achieve settled solutions since unsteady vortex shedding occurs.The HFBM simulations consisted of modeling an isolated propeller with no other bodies in the flow. The flow was uniform and at 0° angle of attack relative to the axis of rotation. Therefore, the flow at each blade was periodic and steady-state when viewed in the fixed-blade reference frame. Only one blade was modeled, as the problem was periodic and thus periodic boundary conditions were applied to the axisymmetric planes.

AFLR (advancing–front, local–reconnection) was used to generate the unstructured mesh. The entire mesh was rotated for unsteady simulations in which one time-step corresponded to one degree of rotation. A time-step study was conducted to ensure the time-step used was small enough to accurately resolve the flow field. The grid was rotated for five revolutions so the force on the blade was settled without any start-up effects.

Computational efficiency could be gained by simulating the propeller as a steady-state computation in the fixed blade reference frame. However, unsteady computations were conducted for purposes of similarity to other CFD simulations in related research. The surface of the blade was divided into sections so the CHEM code could output the total force (viscous and inviscid) vector on each blade element. No wall functions were used, and so the grid near the viscous surface was refined to ensure the boundary layer is captured with a high resolution.The top and bottom surfaces of the blade were each covered with 66 points. A far field size study was conducted to ensure the outer boundary of the computation domain was far enough away to not affect the propeller aerodynamics. The outer boundary was 12 blade lengths away from the propeller blade. The total grid size was 5.6 million elements, and the HFBM simulations were run in a few hours on the Talon super computer at the High Performance Computing Collaboratory of MSU.

#### Blade-Element Momentum Theory

To implement BEMT, a set of lift and drag curves were needed for the NACA 4412. The lift and drag on a 2-D airfoil are functions of angle of attack, Reynolds number, and Mach number. The tip Mach number for the propeller cases was small, <0.32, so compressibility effects were assumed to be negligible.

These low Mach numbers are typical for small-scale propellers due to the low flight speed and small propeller diameter. Some SUAVs have very high propeller rotation speeds causing the flow at the blade tip to be compressible despite the small propeller diameter. In these cases, Mach number can be considered in BEMT. However, for the test cases here it was unnecessary to include compressibility effects as the tip Mach number was low.

To conduct the CFD simulations to make the lift and drag curves, the Mach number was held constant at a moderate value of 0.15. Airfoil simulations covered the range of the Reynolds number experienced by each blade element (10,000-150,000). This range of Reynolds variation can have a significant effect on the airfoil's lift and drag, especially when a turbulence model is used.

A database of lift and drag data for the NACA 4412 airfoil was developed that covered the range of angle of attack and Reynolds number experienced by the blade elements for the propeller cases. For a direct comparison of BEMT to the HFBM simulations, similarity was maintained as much as possible between the 2-D airfoil CFD simulations and the HFBM simulations.

The CHEM code was also used to perform 2-D airfoils simulations with the SST turbulence model. For grid similarity, the same number and distribution of points used on a blade cross section for the HFBM grid was also used on the 2-D airfoil grid that was also made with AFLR. Therefore, the 2-D airfoil grid looked similar to the cross section of the 3-D grid generated for the HFBM mesh. In addition, the boundary layer was captured to a similar resolution as in the HFBM.

The BEMT model was programmed in Mathematica and only took a few minutes to run on a personal computer. Momentum theory was chosen as it is one of the most commonly used methods to calculate the induced velocities for blade-element theory (BET). BEMT is well documented in literature and is easily implemented with an iterative solution procedure. Prandtl's tip and hub loss correction factors were incorporated with the model, and no stall-delay model was used.

#### Analyzing Results

For propellers with high aspect ratio blades operating in conditions with little separation, BEMT was able to closely predict the distribution of thrust along the blade, as the 3-D effects were small. However, as the 3-D effects increased by way of blade geometry or operating conditions, BEMT lost accuracy and thus applicability.

Correction models can be developed for and applied to specific tip geometries and propellers to achieve better agreement. However, these correction models for tip loss, hub loss, stall-delay, and rotational effects have difficulty in being generalized for a wide range of propeller geometries and operating conditions.

Despite these limitations in applicability, BET models are widely used when modeling propellers in CFD as they can be implemented as a computationally efficient steady-state model.

HFBM provided a time-accurate, high-resolution solution for the propeller that considered all 3-D effects. However, HFBM comes at a very high computational cost.

A fine resolution grid is needed to resolve the flow around the propeller. In addition, the problem is time-dependent and restricted to a small time-step to resolve the fast propeller rotation. This high computational cost of HFBM limits its use for many applications despite the high accuracy.

It is worth noting that the 3-D effects of separation and low aspect ratio that cause inaccuracies with BET are not specifically unique to small-scale propellers. Full-scale aircraft propellers, wind turbines, and other propeller or fan applications may also have strong 3-D effects, making BET insufficient. However, small-scale propellers on SUAVs are particularly prone to strong 3-D effects from blade geometry, non-variable pitch blades, and operation in a wide range of flight conditions. The results presented here are intended to show situations in which the 2-D flow assumption of BET breaks down. Therefore, full 3-D CFD was compared directly to BEMT, whose aerodynamic database was developed as similar as possible to the HFBM simulations. This comparison allowed a detailed analysis of the flow field to examine why BEMT loses accuracy in certain flight conditions. The physical accuracy of HFBM to experimental data is not guaranteed despite its consideration of 3-D effects. However, it is beyond the scope here to compare HFBM to experimental data, as HFBM is widely used and accepted in detailed propeller analysis. Other work has compared experimental results, rather than HFBM, to BEMT calculations. In such a case, global thrust quantities were compared rather than thrust distributions along the blade, and similar results were found showing how BEMT is inaccurate in cases with separated flow.#### Propeller-Aircraft Coupling

The main objective of propeller modeling in this SUAV context was propeller-aircraft coupled CFD simulation. Often, the desired information from these coupled CFD simulations are the time-averaged loads on the aircraft that require numerous simulations. To save computation time, a steady-state, computationally efficient momentum source method is desired.Currently, BET is the most accurate and common method on which to base a momentum source model in CFD. When implemented in 3-D CFD, BET does not necessarily need another model to calculate the induced velocities, as the 3-D CFD solution calculates the induction by satisfying the Navier-Stokes equations over the flow domain. Momentum source term models based on BET that are implemented into 3-D CFD are well documented and are currently the most popular way to implement a time-averaged propeller model.

For such models, the magnitude of the source terms are based on 2-D airfoil characteristics and are calculated from the inputs of angle of attack, Reynolds number, and Mach number taken locally in the flow field. Therefore, the source terms are locally coupled to the flow field and adapt as the solution progresses. Due to the local inputs, different flight conditions and interference effects from aircraft couplings are considered in the calculation of the source terms.

Nonetheless, the 2-D flow assumption of BET fails to account for many of the complex 3-D flow characteristics that can significantly affect propeller performance, limiting its accuracy and range of applicability. Fundamentally, the BET assumes the flow over each element to be 2-D in nature.

However, the work here has shown that propeller aerodynamics can be highly 3-D and thus not accurately predicted by 2-D airfoil data. To obtain accurate loads on an aircraft that is affected by the propeller flow, the magnitude of momentum sources must be correct.

So while the momentum source term implementations of BET are locally adaptive to different flow conditions and aircraft couplings, the magnitude of the source terms can have considerable errors when 3-D effects are significant on the propeller, as is often the case for small-scale fixed-blade propellers.

*This article is based on SAE International technical paper 2013-01-2270 by Joseph Carroll and David Marcum, Mississippi State University.*

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