Inflow Models for Blade Element Momentum Theory in Forward Flight

An assignment for the course FLIGHT DYNAMICS AND AEROELASTICITY OF ROTARY-WING AIRCRAFT of Politecnico di Milano

'polarPcolor' function thanks to the contribution of @ECheynet, polarPcolor github page.

Qiuyang Xia

Abstract

In this report, blade element momentum theory is briefly introduced. Momentum theory is utilized to first determine the inflow distribution of the rotor during a forward flight scenario, and then blade element analysis is used to calculate the thrust coefficient of each blade element. The reference blade airfoil is NACA0012, and the blade parameters and flight configurations are taken from a NASA experiment conducted in 1988. Several different inflow models tested in this report including uniform inflow model and various linear inflow model. Results showed the thrust coefficient distribution varies in terms of different inflow model, but correspond to a similar trend.

Method

Uniform Inflow Model Derived from Momentum Analysis

During forward flight, the rotor disk is tilted forward by an angle $\alpha$. The geometric and force analysis is shown in Figure 1:

Figure 1. Glauert's flow model for the momentum analysis of a rotor in forward flight.

where $V_{\infty}$ being infinite distance flow, $v_{i}$ being induced velocity perpendicular to the rotor disk, and $w$ being induced velocity at far wake.

Glauert made the hypothesis of mass inflow:

$$\dot{m} = \rho AU$$

where $\rho$ is the air density, $A$ is the rotor area, and $U$ is the resultant velocity at the disk:

$$U = \sqrt{\left( V_{\infty}\cos\alpha \right)^{2} + \left( V_{\infty}\sin\alpha + v_{i} \right)^{2}} = \sqrt{V_{\infty}^{2} + 2V_{\infty}v_{i}\sin\alpha + v_{i}^{2}}$$

With the conservation of momentum, we can get:

$$\dot{m}\left( V_{\infty}\sin\alpha + w \right) - \dot{m}V_{\infty}\sin\alpha = \dot{m}w = T$$

From the energy conservation, we can get:

$$\frac{1}{2}\dot{m}\left( V_{\infty}\sin\alpha + w \right)^{2} - \frac{1}{2}\dot{m}V_{\infty}^{2}\sin^{2}\alpha = \frac{1}{2}\dot{m}\left( 2V_{\infty}w\sin\alpha + w^{2} \right) = P = T\left( V_{\infty}\sin\alpha + v_{i} \right)$$

From which we can get the relationship between $w$ and $v_{i}$:

$$2V_{\infty}\sin\alpha + w = 2\left( V_{\infty}\sin\alpha + v_{i} \right)$$

$$w = 2v_{i}$$

Thus, total thrust can be expressed as follows:

$$T = 2\dot{m}v_{i} = 2\rho AUv_{i} = 2\rho Av_{i}\sqrt{V_{\infty}^{2} + 2V_{\infty}v_{i}\sin\alpha + v_{i}^{2}}$$

Since hovering induced velocity can be expressed as:

$$v_{h}^{2} = \frac{T}{2\rho A}$$

the induced velocity can be expressed as:

$$v_{i} = \frac{v_{h}^{2}}{\sqrt{\left( V_{\infty}\cos\alpha \right)^{2} + \left( V_{\infty}\sin\alpha + v_{i} \right)^{2}}}$$

Defining the advance ratio and the inflow ratio in forward flight as $\mu$ and $\lambda$ respectively, we have:

$$\mu = \frac{V_{\infty}\cos\alpha}{\Omega R}$$

$$\lambda = \frac{V_{\infty}\sin\alpha + v_{i}}{\Omega R} = \mu\tan\alpha + \lambda_{i}$$

From the equations above, the induced inflow ratio can be expressed as:

$$\lambda_{i} = \frac{\lambda_{h}^{2}}{\sqrt{\mu^{2} + \lambda^{2}}} = \frac{C_{T}}{\sqrt{\mu^{2} + \lambda^{2}}}$$

Thus, we get an implicit expression of inflow ratio:

$$\lambda = \mu\tan\alpha + \frac{C_{T}}{\sqrt{\mu^{2} + \lambda^{2}}}$$

Linear Inflow Model

In-flight experimental observations at higher advancing speed ($\mu > 0.15$) have shown that the longitudinal and lateral inflow variation is approximately linear, which can be expressed as:

$$\lambda_{i} = \lambda_{0}\left( 1 + k_{x}\frac{x}{R} + k_{y}\frac{y}{R} \right) = \lambda_{0}\left( 1 + k_{x}r\cos\psi + k_{y}r\sin\psi \right)$$

In this equation, $\lambda_{0}$ is the mean induced velocity at the center of the rotor as given by the uniform momentum theory. $k_{x}$ and $k_{x}$ can be viewed as weighting factors and represent the deviation of the inflow from the uniform value predicted by the simple momentum theory. $\psi$ is the blade azimuth position angle.

Various researchers suggested different value couples for the coefficients in the linear inflow model, which is summarized in Table 1.

Table 1. Various Estimated Values of First Harmonic Inflow

Author(s)	$k_{x}$	$k_{y}$
Coleman et al. (1945)	$\frac{\tan\chi}{2}$	$0$
Drees (1949)	$\frac{4}{3}\frac{1 - \cos\chi - 1.8\mu^{2}}{\sin\chi}$	$-2\mu$
Payne (1959)	$\frac{4}{3}\frac{\mu}{\lambda\left( 1.2 + \frac{\mu}{\lambda} \right)}$	$0$
White & Blake (1979)	$\sqrt{2}\sin\chi$	$0$
Pitt & Peters (1981)	$\frac{15\pi}{23}\tan\frac{\chi}{2}$	$0$
Howlett (1981)	$\sin^{2}\chi$	$0$

Blade Element Analysis in Forward Flight

A few blade element assumptions are made for the analysis. First, the aerodynamic characteristics of blade sections are independent. And second, the radial component of velocity does not affect lift an drag. Small angles assumption is also utilized in literature, but since the calculation in this report is conducted by PC, this assumption is neglected.

Figure 2 shows a typical blade element for analysis. The $U_{R}$ is the radial component of velocity, which is neglected in this analysis.

Figure 2. Incident velocities and aerodynamic environment at a typical blade element.

$U_{T}$ is the in-plane component of the resultant local flow velocity:

$$U_{T} = \Omega y + \mu\Omega R\sin\psi$$

and $U_{P}$ is the out-of-plane component.

$$U_{P} = \lambda\Omega R + y\dot{\beta} + \mu\Omega R\beta\cos\psi$$

Therefore, the resultant velocity at the blade element is:

$$U = \sqrt{U_{T}^{2} + U_{P}^{2}}$$

The induced angle of attack, or the relative inflow angle $\phi$ at the blade element will be:

$$\phi = \tan^{- 1}\frac{U_{P}}{U_{T}}$$

If the pitch angle at the blade element is $\theta$, the the aerodynamic effective angle of attack $\alpha$ (different from shaft tilt angle) is:

$$\alpha = \theta - \phi$$

If the lift and drag coefficient is a function of effective angle of attack, they can be expressed as follows respectively:

$$C_{l} = C_{l\alpha}\alpha$$

$$C_{d} = C_{d\alpha}\alpha$$

Thus, the resultant incremental lift $dL$ and drag $dD$ per unit span on the blade element are:

$$dL = \frac{1}{2}\rho U^{2}cC_{l}dy$$

$$dD = \frac{1}{2}\rho U^{2}cC_{d}dy$$

where $c$ is the local blade chord.

Referring to Figure 2, the forces perpendicular and parallel to the rotor disk can be resolved respectively as:

$$dF_{z} = dL\cos\phi - dD\sin\phi$$

$$dF_{x} = dL\sin\phi + dD\cos\phi$$

Therefore, the blade element contributions to the thrust, torque, and power of the rotor are:

$$dT = N_{b}dF_{z}$$

$$dQ = N_{b}dF_{x}y$$

$$dP = N_{b}dF_{x}\Omega y$$

where $N_{b}$ is the number of blades.

For conveniently analyze, the non-dimensional quantities are introduced as follows:

$$dC_{t} = F \frac{dT}{\rho A(\Omega R)^{2}}$$

$$dC_{q} = F \frac{dQ}{\rho A(\Omega R)^{2}R}$$

$$dC_{p} = F \frac{dP}{\rho A(\Omega R)^{3}}$$

where $F$ is the Prandtl's tip-loss function, which will be introduced in the following section.

Prandtl's Tip-Loss Function

The formation of tip vortices produces large inflow and local lift reduction which can be accounted for by either defining a simple ratio factor, or by computing tip-loss effects based on a method devised by Prandtl. The tip-loss factor derived by Prandtl can be expressed by $F$:

$$F = \frac{2}{\pi}\cos^{- 1}e^{- f}$$

where $f$ is given in terms of the number of blades and the radial position of the blade element:

$$f = \frac{N_{b}}{2} \frac{1 - r}{r\phi}$$

and the $\phi$ is the induced inflow angle.

$$\phi = \frac{\lambda(r)}{r}$$

Characteristics and parameters derived from literature and used for calculating

The rotor and blade characteristics and performance parameters are taken from an experiment conducted by NASA in 1988, the experiment report was written by Elliott et al.

Table 2. Rotor and blade characteristics

Characteristics	Symbol	Value
Number of blades	$N_{b}$	4
Airfoil		NACA0012, $C_{l\alpha} = 5.73$
Hinge offset		0.0508 m
Root cutout		0.2096 m
Pitch-flap coupling		0
Linear twist	$\theta_{tw}$	-8 deg
Radius	$R$	0.8606 m
Root chord	$c$	0.0660 m

Table 3. Rotor performance parameters

Parameters	Symbol	Value
Drag coefficient	$C_{D}$	0.0002
Shaft tilt angle	$\alpha_{s}$	-3 deg
Coning	$\beta_{0}$	1.5
Infinite inflow ratio	$\mu_{\inf}$	0.149
Thrust coefficient	$C_{T}$	0.0063
Infinite inflow velocity	$V_{\inf}$	28.5002 m/s
Blade tip velocity	$V_{tip}$	190.2866 m/s

It's worth noticing that the NASA report nominated rotor control values:

$$\theta = A_{0} - A_{1}\cos\psi - B_{1}\sin\psi$$

In the report, $A_{0} = 9.37,A_{1} = - 1.11,B_{1} = 3.23$. However, subsequent blade element analysis proved that these parameters caused unbalanced rotor load in the lateral direction. Thus, in this report, another set of parameters were derived from the Master's thesis of Güner, 2016. where:

$$\theta = \theta_{0} - \theta_{1c}\cos\psi - \theta_{1s}\sin\psi$$

and $\theta_{0} = 6.26,\theta_{1c} = - 2.08,\theta_{1s} = 1.96$.

The linear twisted blade can be expressed as:

$$\theta(r) = \theta_{75} + (r - 0.75)\theta_{tw}$$

The rotor flapping in forward flight is commonly expressed as a first order harmonics:

$$\beta = \beta_{0} + \beta_{1c}\cos\psi + \beta_{1s}\sin\psi$$

However, in the NASA report, the flapping angle was set to constant with a coning angle $\beta_{0} = 1.5{^\circ}$, resulting:

$$\beta = \beta_{0}$$

$$\dot{\beta} = 0$$

Results

Inflow Model Comparison

Figure 3 plotted the measured data from the NASA report and the multiple inflow model mentioned in the previous section. It is worth noticing that the positive direction defined by the NASA report is vertical to the rotor disk plane upwards, which is the opposite to the inflow models. Thus, the NASA data plotted in Figure 3 is multiplied by -1 to correspond to the inflow models.

a) Inflow in longitudinal 0° - 180° direction, positive 0°.

b) Inflow in lateral 90° - 270° direction, positive 90°, missing data for 270° direction in NASA measurement.

Figure 3. Measured and modeled inflow ratio at $\mu = 0.149, C_{T} = 0.0064,\alpha_{s}= - 3^{\circ}$.

From Figure 3 we can observe that the Coleman linear inflow model had the most gentle slope in the longitudinal direction, while the Pitt & Peters linear inflow model had the most aggressive slope. This was the result of their choosing in the $k_{x}$ expression. Drees linear inflow model were most approximate to the measured data.

Regarding to the lateral direction, since the above mentioned model all assumed that $k_{x} = 0$ except for Drees, their curve in the lateral plot coincident with the x-axis.

Theta Configuration

Due to the pitch control configurations, the theta value across the blade span and azimuth angle differs in each position. Figure 4 showed the corresponding theta value in each position in degrees.

Figure 4. Theta Configuration (deg)

Blade Element Analysis of Thrust Coefficient

In this section, the distribution of inflow ratio, effective angle of attack (AoA), and thrust coefficient among the blade span and azimuth position is plotted. Also, define the "rotor balancing ratio" as:

$$B = \frac{\int_{0}^{\pi}{\int_{}^{}{dC_{T}}}}{\int_{\pi}^{2\pi}{\int_{}^{}{dC_{T}}}}$$

which evaluates the thrust balancing of advancing side and retreating side of the rotor.

From Figure 5 to Figure 11, different inflow models were presented. But we can acquire some mutual phenomenon among them. The reverse flow region was around 0° to 180° of azimuth position within the blade span of 0.4, where the thrust coefficient also displayed the minimum value. The largest value of thrust coefficient always appeared near the blade tip position, although the azimuth position may differ.