Advertisement

Aerospace Systems

, Volume 1, Issue 1, pp 31–37 | Cite as

Ground effect analysis for unmanned lifting rotor through robust finite state inflow model based on adjoint theorem

  • Jianzhe HuangEmail author
  • Jinshuo Hu
Original Paper
  • 173 Downloads

Abstract

When helicopter is climbing from the ground or flying close to a solid surface, the flow around the lifting rotor can be strongly affected such that it will become time varying. To control the helicopter when the ground effect presents, a fast and accurate simulation technique is required. Traditional finite state inflow model with ground effect still need numerical integration since it cannot directly give prediction below the disk, and the computation of wake is necessary due to its interaction with ground. To improve the efficiency of the model, the adjoint theorem is adopted such that the wake can be solved with closed form equations. The result in hover condition is validated with Hayden model which is correlated well with the experimental data.

Keywords

Wake modeling Finite-state inflow model Adjoint Ground effect 

1 Background

In 1960, Fradenburgh [1] showed that helicopter could gain more payloads due to the ground effect, but the ground effect machine could not compete with helicopter for the many missions and such a machine had to be restricted to very specialized assignments. NASA [2] reported that rotors in forward flight within ground effects could lead a decrease in induced flow and forward speed. At very low height above the ground, the relationship between power requirements and speed became nonlinear, and stability problems were observed due to such an effect. Curtiss et al. [3] experimentally studied the aerodynamic characteristics of an isolated rotor at low advance ratio near the ground. Cerbe and Reichert [4] investigated the optimization of the special runway Cat. A takeoff, and CDP (Critical Decision Point) was determined by flight tests. Graber et al. [5] applied a new free-wake aerodynamic model for a rotor hovering close to the ground, and the thrust coefficient correlated well with the experimental data. Itoga et al. [6] combined a free-wake method with a panel method to make sure the blade flapping motions to be consistent with the deformed wake geometry when a rotor was in I.G.E hover condition. Mahony and Hamel [7] studied the control scheme for a scale model autonomous helicopter during takeoff and landing manoeuvers, and the influence on thrust for ground effect was considered by introducing a coefficient which varies with the height of the helicopter above the ground. Yu and Peters [8] introduced a ground source rotor model which was coupled with finite-state inflow model of main rotor to model the ground effect of helicopter in hover condition, and simulation results gave good correlations compared with the validated empirical models. Roy et al. [9] designed a nonlinear robust control to control the altitude of a small helicopter for hover near ground surface, and the reduction factor of induced velocity due to ground effect was considered. Garrick et al. [10] applied CFD approach to simulate a descending rotorcraft, which transited from OGE (Out of Ground Effect) into IGE (In Ground Effect). To guarantee the flight safety issue for helicopter hovering at low height, the effects of confined area geometry on the aerodynamic performance was studied experimentally in [11]. Xin, Chen and Li [12] predicted ground influence on rotor flow field through constant volume rectification technique, and the development of discrete error was inhibited using a so called CB3D (Center difference and Backward difference 3rd-order scheme with numerical Dissipation) algorithm with strengthened stability. Hooi [13, 14] applied a ring-source potential-flow model to represent the ground effect for real-time use, and such a model was validated with the experimental data. Sanchez-Cuevas et al. [15] analyzed multirotors flying close to the ground, and interaction of the rotor airflow with ground surface was discussed.

To design a controller for helicopter of which the flight mode could change with time, a fast computational model is required. Peters and He [16] developed a generalized finite-state inflow model with an arbitrary number of states, and the computed normal component of flow on the disk had been verified with test data. To deal with ground effect, the aforementioned researches have revealed many important characteristics of the flow field when the rotor is flying close to the ground. For empirical models, it works well for hover conditions. But for more advanced flight mode such as forward flight, it needs a large number of flight data and curve fitting work, and the transient process cannot be modelled. For most of the numerical approaches such as CFD and vortex panel, it is difficult to fulfill the requirement of a close-loop control for which the loading and rotor status change with time. Yu [8] developed the finite-state inflow model with ground effect, but it cannot give the close-form solution for induced velocity below disk, and the wake of main rotor is part of inflow of the ground rotor. To improve the model which Yu developed, adjoint theorem will be introduced to compute the flow field below the main rotor. Adjoint variables with time delay will be derived and they will be expressed in a convolution form such that numerical time-marching process to calculate such variables is avoided. The number of states will be discussed to make sure the boundary condition for load determination of ground rotor satisfies. With such a technique, the thrust ratio for IGE (In Ground Effect) and OGE (Out Ground Effect) at constant power for hover condition is verified with famous Hayden model. Then the transient response of lifting rotor at different height is simulated with the constant collective pitch.

2 Flow below the disk

Within the wake, the vortex sheets exist and the potential flow model is no longer valid to give predictions. Thus, adjoint theorem has to be introduced and it is described as follows

For the region shaded by the wake which is shown in Fig. 1, traditional Morillo–Duffy model or Nowak-He model cannot be used to calculate the flow field in such a region. Instead, to obtain the induced velocity at a certain location within the wake such as point A, one need to know the information for points B, C and D, which are either on the disk or above the disk. r measures radial distance from the centerline of rotor disk, and ψ is the azimuth angle. ξ is the coordinate along the streamline, and it is positive below the disk. Thus, the adjoint theorem used to calculate the induced velocity at point A can be given with a mathematical expression in Eq. (1). The first term is the induced velocity at point B with time delay ξ0/VT, and the last two terms are the adjoint induced velocity at point C with time delay ξ0/VT and point C, respectively. VT is the total velocity at the rotor disk and \( V_{\text{T}} \, = \,|{\mathbf{V}}_{\infty } \, + \,\overline{{{\mathbf{v}}_{z} }} | \), where \( {\mathbf{V}}_{\infty } \, = \,V_{\infty x} {\mathbf{i}}\, + \,V_{\infty y} {\mathbf{j}} \) is the freestream velocity and \( {\bar{\mathbf{v}}}_{\text{z}} \) is the average axial induced velocity at the disk.
Fig. 1

Flow calculation for wake based on adjoint theorem

$$ v\left( {r_{0} ,\psi_{0} ,\xi_{0} ,t} \right) = \vec{v}\left( {r_{0} ,\psi_{0} ,0,t - \frac{{\xi_{0} }}{{V_{\text{T}} }}} \right) + \vec{v}^{*} \left( {r_{0} ,\psi_{0} + \pi ,0,t - \frac{{\xi_{0} }}{{V_{\text{T}} }}} \right) - \vec{v}^{*} \left( {r_{0} ,\psi_{0} + \pi , - \xi_{0} ,t} \right). $$
(1)
As is discussed in [17], Nowak–He model is accurate on the rotor disk and Morillo–Duffy is valid off the disk. Therefore, the first two terms in Eq. (1) can be calculated using Nowak–He model. For Morillo–Duffy model, it has
$$ R{\mathbf{M}}\left\{ {\dot{a}_{n}^{m} } \right\} + {\mathbf{DVL}}^{ - 1} {\mathbf{M}}\left\{ {a_{n}^{m} } \right\} = {\mathbf{D}}\left\{ {\tau_{n}^{m} } \right\}, $$
(2)
where M is the mass matrix, L is the influence coefficient matrix, and D is the damping matrix. All of the detailed expressions for the aforementioned matrices can be found in [18], and M = L when it is in hover. R is the radius of the rotor. The flow related matrix V is described in Eq. (3).
$$ {\mathbf{V}} = \left[ {\begin{array}{*{20}c} {V_{\text{T}} } & 0 & 0 & 0 \\ 0 & V & \vdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & V \\ \end{array} } \right] $$
(3)
where
$$ V_{\text{T}} = \left| {{\mathbf{V}}_{\infty } + {\bar{\mathbf{v}}}_{z} } \right| $$
$$ V = \frac{{{\text{d}}(\bar{v}_{\text{z}} V_{\text{T}} )}}{{{\text{d}}(\bar{v}_{\text{z}} )}} $$
$$ \bar{v}_{\text{z}} = \frac{2}{\sqrt 3 }\left\{ {\begin{array}{*{20}c} 1 & 0 & \cdots & 0 \\ \end{array} } \right\}{\mathbf{L}}^{ - 1} {\mathbf{M}}\left\{ {a_{n}^{m} } \right\} $$
\( \left\{ {\tau_{n}^{m} } \right\} \) is the pressure input and it contains two parts: m + n equals an odd number and m + n equals an even number. The structure of such a vector is expressed as \( \left\{ {\tau_{n}^{m} } \right\} = \{ \tau_{1}^{0} ,\tau_{3}^{0} , \ldots ,\tau_{2}^{1} ,\tau_{4}^{1} , \ldots ,\tau_{0}^{0} ,\tau_{2}^{0} , \ldots \} \). For the element which m + n equals an even number, \( \tau_{n}^{m} \) is zero. Otherwise, it is calculated as in Eq. (4).
$$ \begin{aligned} \tau_{n}^{0} & = \frac{1}{2\pi }\sum\limits_{q = 1}^{Q} {\int_{0}^{1} {\frac{{L_{q} }}{2\rho R}J_{0} (0)\phi_{n}^{0} (r){\text{d}}r} } ,{\text{ for }}n{\text{ is odd;}} \\ \tau_{n}^{m} & = \frac{1}{\pi }\sum\limits_{q = 1}^{Q} {\int_{0}^{1} {\frac{{L_{q} }}{2\rho R}J_{0} \left( {m\overline{b} /r} \right)\phi_{n}^{m} (r){\text{d}}r\cos \left( {m\psi_{q} } \right),{\text{for }}m = 1,2, \cdots {\text{ and }}n = m + 1,m + 3, \cdots ,} } \\ \end{aligned} $$
(4)
where ρ is air density, Q is number of blade, ψq is azimuth angle of the qth blade and ψq = Ωt + 2π/Q (q–1), \( \bar{b} \) is the non-dimensional semi-chord and \( \bar{b} = c/2R \). c and Ω are the chord of the blade and rotation speed, respectively. \( \phi_{n}^{m} (r) \) is a function which is expressed as
$$ \phi_{n}^{m} (r) = \frac{1}{{\sqrt {1 - r^{2} } }}P_{n}^{m} \left( {\sqrt {1 - r^{2} } } \right), $$
(5)
where \( P_{n}^{m} \) is the Legendre function of the first kind. The sectional lift per unit length for the qth blade can be calculated with Eq. (6).
$$ \begin{aligned} L_{q} (r) = & \frac{\rho ac}{2}\left[ {\left( {\varOmega Rr + \mu \varOmega R\sin \psi_{q} } \right)^{2} \theta - \left( {\varOmega Rr + \mu \varOmega R\sin \psi_{q} } \right)} \right. \\ & \left. {\left( {\eta_{s} \varOmega R + v + \dot{\beta }_{q} Rr + \mu \varOmega R\cos \psi_{q} \beta_{q} } \right)} \right] \\ \end{aligned} $$
(6)
where μ is the advanced ratio and ηs is the climb rate; a is the slope of the lift coefficient CL; v is the induced velocity; βq is the flapping angle of the qth blade; θ is the pitch angle and θ = θ0+ θccosψq + θssinψq.
Through the change of variables, the Morillo–Duffy variables \( \left\{ {a_{n}^{m} } \right\} \) can be transferred into Nowak–He variables \( \left\{ {\alpha_{n}^{m} } \right\} \) as is given in Eq. (7).
$$ \left\{ {\alpha_{n}^{m} } \right\}_{{m + n = {\text{odd}}}} = {\mathbf{A}}\left\{ {a_{n}^{m} } \right\}_{{m + n = {\text{odd}}}} {\text{ and }}\left\{ {\alpha_{n}^{m} } \right\}_{{m + n = {\text{even}}}} = \left\{ {a_{n}^{m} } \right\}_{{m + n = {\text{even}}}} $$
(7)
where
$$ \left[ {A_{nj}^{m} } \right] = \frac{{( - 1)^{{\frac{n + j - 2r}{2}}} 2\sqrt {2n + 1} \sqrt {2j + 1} }}{{\sqrt {H_{n}^{r} } \sqrt {H_{j}^{r} } (n + j)(n + j + 2)\left[ {(n - j)^{2} - 1} \right]}}, $$
$$ H_{n}^{m} = \frac{(n + m - 1)!!(n - m - 1)!!}{(n + m)!!(n - m)!!}. $$
For the adjoint variables based on Morillo–Duffy model \( \left\{ {\Delta_{n}^{m} } \right\} \), it gives
$$ - R{\mathbf{D}}^{ - 1} {\mathbf {M}}\mathop {\left\{ {\Delta_{n}^{m} } \right\}}\limits^{ \cdot } + {\mathbf{VL}}^{ - 1} {\mathbf{M}}\left\{ {\Delta_{n}^{m} } \right\} = \left[ {\begin{array}{*{20}c} \ddots & {} & {} \\ {} & {\left( { - 1} \right)^{n + 1} } & {} \\ {} & {} & \ddots \\ \end{array} } \right]\left\{ {\tau_{n}^{m} } \right\}. $$
(8)
Then the adjoint variables based on Nowak-He model \( \left\{ {\varLambda_{n}^{m} } \right\} \) can be obtained using the similar transformation in Eq. (9).
$$ \left\{ {\varLambda_{n}^{m} } \right\}_{{m + n = {\text{odd}}}} = {\mathbf{A}}\left\{ {\Delta_{n}^{m} } \right\}_{{m + n = {\text{odd}}}} {\text{ and }}\left\{ {\varLambda_{n}^{m} } \right\}_{{m + n = {\text{even}}}} = \left\{ {\Delta_{n}^{m} } \right\}_{{m + n = {\text{even}}}} . $$
(9)
Since the Nowak–He model give better predictions on the disk and Morillo–Duffy model preforms well off the disk, the first two terms in Eq. (1) will be computed based on Nowak–He model. But for the adjoint velocity with time delay, the time-delayed adjoint variables are required. To improve the computational efficiency, one has to seek closed form representation for such time-delayed adjoint variables. Define
$$ {\mathbf{S}} = R^{ - 1} {\mathbf{M}}^{ - 1} {\mathbf{DVL}}^{ - 1} {\mathbf{M}}\;{\text{and}}\;{\mathbf{F}} = R^{ - 1} {\mathbf{M}}^{ - 1} {\mathbf{D}}\left[ {\begin{array}{*{20}c} \ddots & {} & {} \\ {} & {\left( { - 1} \right)^{n + 1} } & {} \\ {} & {} & \ddots \\ \end{array} } \right]. $$
(10)
Then one can rewrite Eq. (8) into
$$ - \mathop {\left\{ {\Delta_{n}^{m} } \right\}}\limits^{ \cdot } + {\mathbf{S}}\left\{ {\Delta_{n}^{m} } \right\} = {\mathbf{F}}\left\{ {\tau_{n}^{m} } \right\} $$
(11)
The eigenvalues {ηi} and eigenvector φ of matrix A has the following relationship
$$ {\varvec{\upvarphi }}^{ - 1} {\mathbf{S}}{\varvec{\upvarphi }} = \left[ {\begin{array}{*{20}c} {\eta_{1} } & {} & {} \\ {} & \ddots & {} \\ {} & {} & {\eta_{i} } \\ \end{array} } \right] $$
(12)
The closed-form solution of adjoint variables with time delay \( \varsigma \) is given as
$$ \left\{ {\Delta_{n}^{m} (t - \varsigma )} \right\} = \int_{t - \varsigma }^{t} {{\boldsymbol{\upvarphi W\upvarphi }}^{ - 1} {\mathbf{F}}\left\{ {\tau_{n}^{m} (\alpha )} \right\}d\alpha } $$
(13)
where
$$ {\mathbf{W}} = \left[ {\begin{array}{*{20}c} {{\text{e}}^{{ - \eta_{1} (\alpha - t + \varsigma )}} } & {} & {} \\ {} & \ddots & {} \\ {} & {} & {{\text{e}}^{{ - \eta_{i} (\alpha - t + \varsigma )}} } \\ \end{array} } \right] $$
For the adjoint variables without time delay, it has \( \varsigma = 0 \). According to Eq. (13), such adjoint variables will be always zero. Then the third term in Eq. (1) can be dropped, and it deduces to Eq. (14) as follows.
$$ v\left( {r_{0} ,\psi_{0} ,\xi_{0} ,t} \right) = v\left( {r_{0} ,\psi_{0} ,0,t - \frac{{\xi_{0} }}{{V_{\text{T}} }}} \right) + v^{*} \left( {r_{0} ,\psi_{0} + \pi ,0,t - \frac{{\xi_{0} }}{{V_{\text{T}} }}} \right). $$
(14)
For Morillo–Duffy model, the induced velocity and adjoint velocity on and above the disk are given in Eqs. (15) and (16), respectively.
$$ v_{MD} = \sum\limits_{m = 0}^{\infty } {\sum\limits_{n = m}^{\infty } {a_{n}^{m} P_{n}^{m} (\nu )Q_{n}^{m} (i\eta )\cos (m\psi )} } $$
(15)
$$ v_{MD}^{*} = \sum\limits_{m = 0}^{\infty } {\sum\limits_{n = m}^{\infty } {\Delta_{n}^{m} P_{n}^{m} (\nu )Q_{n}^{m} (i\eta )\cos (m\psi )} } $$
(16)
And the induced velocity and adjoint velocity based on Nowak-He model is expressed as
$$ v_{NH} = \sum\limits_{{m + n\; = \;{\text{odd}}}} {\alpha_{n}^{m} \frac{{P_{n}^{m} (\nu )}}{\nu }Q_{m + 1}^{m} (i\eta )\cos (m\psi )} + \sum\limits_{{m + n\; = \;{\text{even}}}} {\alpha_{n}^{m} P_{n}^{m} (\nu )Q_{m + 1}^{m} (i\eta )\cos (m\psi )} . $$
(17)
$$ v_{NH}^{*} = \sum\limits_{m + n = odd} {\varLambda_{n}^{m} \frac{{P_{n}^{m} (\nu )}}{\nu }Q_{m + 1}^{m} (i\eta )\cos (m\psi )} + \sum\limits_{m + n = even} {\varLambda_{n}^{m} P_{n}^{m} (\nu )Q_{m + 1}^{m} (i\eta )\cos (m\psi )} $$
(18)

3 Ground effect modeling

For helicopter flying at height h less than 2R, ground effect will be considered and it is modelled by placing a mass source rotor on the ground where the wake intersects. Figure 2 demonstrates that the ground effect is modeled by placing a ground rotor at the intersection surface of the wake and the ground. For such a ground rotor, it has
$$ R{\mathbf{M}}\left\{ {\dot{c}_{n}^{m} } \right\} + {\mathbf{DV}}_{{\mathbf{L}}} {\mathbf{L}}^{ - 1} {\mathbf{M}}\left\{ {c_{n}^{m} } \right\} = {\mathbf{D}}\left\{ {\varpi_{n}^{m} } \right\}, $$
(19)
where
$$ {\mathbf{V}}_{{\mathbf{L}}} \triangleq \left[ {\begin{array}{*{20}c} {V_{2} } & 0 & 0 & 0 \\ 0 & {V_{2} } & \vdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & {V_{2} } \\ \end{array} } \right] $$
$$ V_{2} = \frac{{d(\bar{v}_{22} V_{T2} )}}{{d(\bar{v}_{22} )}} $$
$$ V_{{{\text{T}}2}} = \left| {{\mathbf{V}}_{\infty } + {\bar{\mathbf{v}}}_{12} + {\bar{\mathbf{v}}}_{22} } \right|, $$
Fig. 2

Schematic of a lifting rotor in ground effect

\( {\bar{\mathbf{v}}}_{22} \) is the average velocity induced by the ground rotor on the ground rotor, and \( {\bar{\mathbf{v}}}_{12} \) is the average velocity induced by the main rotor on the ground rotor. For \( {\bar{\mathbf{v}}}_{12} \), the adjoint theorem in Eq. (14) should be applied to calculate the induced velocity below the main rotor. Since flow of the main rotor is also influenced by the ground rotor, the flow related matrix V for main rotor in Eq. (3) can be rewritten as
$$ {\mathbf{V}} \triangleq \left[ {\begin{array}{*{20}c} {V_{T1} } & 0 & 0 & 0 \\ 0 & {V_{1} } & \vdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & {V_{1} } \\ \end{array} } \right] $$
(20)
where
$$ V_{{{\text{T}}1}} = \left| {{\mathbf{V}}_{\infty } + {\bar{\mathbf{v}}}_{21} + {\bar{\mathbf{v}}}_{11} } \right| $$
$$ V_{1} = \frac{{d(\bar{v}_{11} V_{T1} )}}{{d(\bar{v}_{11} )}} $$

\( {\bar{\mathbf{v}}}_{11} \) is the average velocity induced by the main rotor on the main rotor, and \( {\bar{\mathbf{v}}}_{21} \) is the average velocity induced by the ground rotor on the main rotor.

Because the ground rotor is a fake one and it has no blade, no actual lift will be produced by such a ground rotor. From the physical point of view, the pressure of the main rotor and the ground on the ground rotor should be identical, and velocity induced by the main rotor plus that induced by the ground rotor on the ground rotor should be zero. However, those two boundary conditions can be fulfilled simultaneously. According to [8], it gives more accurate solution with the boundary condition for pressure. Therefore, it gives
$$ \left\{ {\varpi_{n}^{m} } \right\} = \left[ {G_{nj}^{m} } \right]^{T} \left\{ {\tau_{j}^{m} } \right\},n = m,m + 2,m + 4, \ldots \;{\text{and}}\;j = m + 1,m + 3,m + 5, \ldots $$
(21)
where
$$ \left[ {G_{nj}^{m} } \right] = \int_{0}^{1} {P_{j}^{m} (\nu )P_{n}^{m} (\nu_{2} )Q_{n}^{m} (i\eta_{2} ){\text{d}}\nu } . $$

4 Numerical studies

To examine the effectiveness of such a model, rotor parameters for Bo-105 helicopter will be used and they are tabulated in Table 1. To simulate the flow field in the time domain, one needs all the parameters of the lifting rotor.
Table 1

Physical parameters of Bo-105

Parameter

Description

Value

Unit

N

Number of blades

4

R

Rotor radius

4.91

m

c

Blade chord

0.27

m

I β

Blade moment of inertia

231.7

Kgm2

a

Lift slope of the rotor blades

6.133

1/rad

θ tw

Linear blade twist

− 0.14

rad

Ω

Rotor rotational speed

44.4

rad/s

For helicopter in hover condition, V = 0. Then based on momentum theory, the thrust coefficient is obtained as
$$ C_{\text{T}} = 2\bar{\nu }^{2} \left( {\varOmega R} \right)^{ - 2} $$
(22)
Then one can give an approximate collective pitch angle for a designed thrust (T = 25,000 N and CT = 0.00577) which is given in Eq. (23), and no twist of the blade is assumed. For the actual lifting rotor which the twist of blade is nonzero, such a collective pitch angle will produce a different thrust but around the desired one. Herein, since the control system is not considered to adjust the pitch of lifting rotor, the helicopter will be assumed to balance itself by altering its weight for simplicity.
$$ \theta_{0} = \frac{{6\pi RC_{\text{T}} }}{abc} + \frac{2}{3}\bar{\nu }. $$
(23)
For the source rotor placing on the ground, lift cannot be obtained since there is no blade for such an image rotor. Equation (21) gives the transformation to get the input load for ground rotor from \( \left\{ {\tau_{n}^{m} } \right\} \) of the main rotor. But the boundary conditions of Pmain/Pground = 1 at the ground rotor should be satisfied. To study the state sensitiveness of the model for such a boundary condition, one start from the simplest one which only 2 states are contained (\( \tau_{1}^{0} \) and \( \tau_{0}^{0} \)). For only two states, the transformation matrix [G] equals [− 0.7461]. 29 sets of random number of \( \tau_{1}^{0} \) and \( \tau_{0}^{0} \) are created, and the pressure ratio is computed for each set which is plotted in Fig. 3. It can be found that for such a simple model, the ratio is stable but it stays around 1.2. By increasing the states to 8, the ratio stays around 1 for most of the cases, but it can have very irrational number which is not always trustable. It has same situation for model with 18 states. When the number of states increases to 32, the ratio becomes very stable and it is always a value very close to 1 which indicates that the boundary condition can be always satisfied. To compute the induced velocity, 21 of 32 states will be used to increase the efficiency and the accuracy for predicting the flow field is still guaranteed. For more details, it can be found in [17]. In Table 2, the states to be used to calculate the induced velocity are tabulated.
Fig. 3

Comparisons of pressure ratio for different number of states with 29 random cases at h = 0.5R for Bo-105 rotor in hover

Table 2

Illustration for 21 states

 

m

n

m + n = odd

0

1

 

0

3

 

0

5

 

1

2

 

1

4

 

2

3

 

2

5

 

3

4

 

4

5

m + n = even

0

0

 

0

2

 

0

4

 

1

1

 

1

3

 

1

5

 

2

2

 

2

4

 

3

3

 

3

5

 

4

4

 

5

5

In Fig. 4, the average induced velocities on the main rotor and ground rotor for helicopter hovering at height h = 0.5R are illustrated for t < 20 s. The collective pitch angle is set to be 7.993°, longitudinal and lateral cyclic pitch angles are both zero. The initial conditions are given by assuming the helicopter reaches steady state for hovering at a height where no ground effect exists, and it suddenly drops down to h = 0.8R. On the main rotor which is shown in Fig. 4a, the average velocities induced by the main rotor (solid line) and ground rotor (dashed line) are plotted separately. Since it takes time for flow induced by the main rotor arrive the ground, the average velocity on the main rotor induced by the ground rotor is zero initially, and it gradually goes to be negative as time increases. For the average velocity on the ground rotor induced by the main rotor which is denoted by dash curve in Fig. 4b, it is zero at t = 0 as well and it becomes about 1.71 times larger than the average velocity on the main rotor induced by the main rotor at steady state. Such a velocity is the inflow of the ground rotor, and the magnitude of the average induced velocity on the ground rotor which is induced by the ground rotor increases as it grows. At the steady state, the total induced velocity is computed to be 10.264 m/s, and the thrust is 21913 N. To get the same thrust, the average induced velocity for such a Bo-105 rotor without ground effect will be 11.817 m/s. Therefore, the ratio of power required for rotor with ground effect to that without ground effect will be calculated to be 0.8686. For the Hayden model, that value for rotor at height of 0.8 radii is known as 0.8132, the difference is below 7%.
Fig. 4

Average induced velocity for the main and ground rotor in hover condition at height of h = 0.5R: a main rotor and b ground rotor

5 Conclusions

In this paper, a model for lifting rotor with ground effect has been obtained. The inflow of the main rotor and ground rotor was coupled. Since the ground rotor is in the wake of the main rotor, adjoint theorem was adopted to compute the induced velocity for wake region with a rigorous form. Without numerical integration, the computational efficiency can be greatly improved. For the force input of the ground rotor, it has to be transferred from the loading of the main rotor since it is no blade for ground rotor. The optimum set of states has been determined, with which the boundary condition for pressure can be guaranteed. The transient response for Bo-105 rotor at height of 0.8 radii in the hover condition was illustrated, and result was verified by the famous Hayden model.

In the future, the proposed model will be verified with a large scale of numerical simulations, so that such a model can be widely spread in the research and industrial fields. The model for lifting rotor with partial ground effect can also be derived, and the model for lifting rotor in the forward flight condition can be extended. With these improvements, such a finite-state inflow model can be more robust to give prediction for lifting rotor in most of the flight conditions.

References

  1. 1.
    Fradenburgh EA (1960) The helicopter and the ground effect machine. J Helicopter Soc 5(4):24–33CrossRefGoogle Scholar
  2. 2.
    Heyson HH (1977) Theoretical study of the effect of ground proximity on the induced efficiency of helicopter rotors. NASA-TM-X-71951, p 90Google Scholar
  3. 3.
    Curtiss HC, Sun M, Putman WF, Hanker EJ (1984) Rotor aerodynamics in ground effect at low advance ratios. J Helicopter Soc 29(1):48–55CrossRefGoogle Scholar
  4. 4.
    Cerbe T, Reichert G (1989) Optimization of helicopter takeoff and landing. J Aircr 26(10):925–931CrossRefGoogle Scholar
  5. 5.
    Graber A, Rosen A, Seginer A (1991) An investigation of a hovering rotor in ground effect. Aeronaut J 95(945):161–169Google Scholar
  6. 6.
    Itoga N, Nagashima T, Yoshizawa Y, Prasad JVR (2000) Numerical method for predicting I.G.E. hover performance of a lifting rotor. Trans Jpn Soc Aeronaut Space Sci 43(43):122–129CrossRefGoogle Scholar
  7. 7.
    Mahony R, Hamel T (2001) Adaptive compensation of aerodynamic effects during takeoff and landing manoeuvres for a scale model autonomous helicopter. Eur J Control 7(1):43–57CrossRefGoogle Scholar
  8. 8.
    Yu K, Peters DA (2005) Nonlinear state-space modeling of dynamic ground effect. J Am Helicopter Soc 50(3):259–268CrossRefGoogle Scholar
  9. 9.
    Roy TK, Garratt M, Pota HR, Samal MK (2012) Robust altitude control for a small helicopter by considering the ground effect compensation. In: The 10th World Congress on intelligent control and automation, Beijing, China, July 6–8 2012Google Scholar
  10. 10.
    Garrick DP, Guntupalli K, Rajagopalan RG (2013) Simulation of landing maneuvers of rotorcraft in brownout conditions. n: AIAA aviation, Los Angeles, USA, August 12–14, 2013Google Scholar
  11. 11.
    Iboshi N, Itoga N, Prasad JVR, Sankar LN (2014) Ground effect of a rotor hovering above a confined area. Front Aerosp Eng 3(1):7–16CrossRefGoogle Scholar
  12. 12.
    Xin J, Chen RL, Li P (2015) Time-stepping free-wake methodology for rotor flow field simulation in ground effect. Aircr Eng Aerosp Technol An Int J 87(5):418–426CrossRefGoogle Scholar
  13. 13.
    Hooi CG, Lagor FD, Paley DA (2016) Height estimation and control of rotorcraft in ground effect using spatially distributed pressure sensing. J Am Helicopter Soc 61(4):1–14CrossRefGoogle Scholar
  14. 14.
    Hooi CG, Lagor FD, Paley DA (2015) Flow sensing for height estimation and control of a rotor in ground effect: modeling and experimental results. In: The AHS 71st annual forum, Virginal Beach, USA, May 5–7, 2015Google Scholar
  15. 15.
    Sanchez-Cuevas P, Heredia G, Ollero A (2017) Characterization of the aerodynamic ground effect and its influence in multirotor control. Int J Aerosp Eng 2017:17 (article ID: 1823056)Google Scholar
  16. 16.
    Peters DA, Boyd DD, He CJ (1989) A finite-state induced-flow model for rotors in hover and forward flight. J Am Helicopter Soc 34(4):5–17CrossRefGoogle Scholar
  17. 17.
    Huang JZ, Morgan N, Peters DA, Prasad JVR (2014) Converged velocity field for rotors by a blended potential flow method. In: Proceedings of the 70th annual forum of the american helicopter society, Montreal, Canada, 20–22 May 2014Google Scholar
  18. 18.
    Huang J (2015) Potential-flow inflow model including wake distortion and contraction. PhD Thesis, Washington University in St. Louis, USA, 2015Google Scholar

Copyright information

© Shanghai Jiao Tong University 2018

Authors and Affiliations

  1. 1.Harbin Engineering UniversityHarbinChina
  2. 2.Nanjing University of Aeronautics and AstronauticsNanjingChina

Personalised recommendations