The build topology of the CRM model has been checked and corrected to ensure air tightness on the model surfaces. After this procedure a hexahedral mesh was generated for the full model. A structured mapped blocking approach with appropriate splits and inclusion of O-grids was used to better capture the boundary layer regions on the airplane surfaces.

The blocks initially generated, were transformed through rotations and translations to generate hexahedral unstructured meshes according to flight conditions, i.e. airplane attitude and closeness to the ground. The boundary conditions on the ground were implemented as a moving wall with direction and velocity magnitude of incoming flow and were resolved with inclusion of H-grid layers with appropriate wall distance (Y+<1).

The initial meshes were generated for different altitudes above the ground, i.e. h = 4 c ¯ , h = 2 c ¯ , h = 1 c ¯ , and h = 0.5 c ¯ . At each altitude h, the grid was adapted for a number of different angles of attack settings α = 4 ∘ , 8 ∘ , 12 ∘ . At altitude h = 0.5 c ¯ additionally a number of bank angle settings was considered ϕ = 4 ∘ , 8 ∘ , 12 ∘ with additional adaptation of the grid. Figure 2 shows different CRM attitudes at h = 0.5 c ¯ (different colors are used to highlight different aerodynamic surfaces of the model). The blue mesh in the bottom represents the ground.

The numerical simulations were carried out within reasonable accuracy of a grid between coarse to medium, i.e. about 10 million cells for a full body configuration. This seems suitable for our purpose here to evaluate ground effect.

The Navier-Stokes equations governing incompressible fluid flow are:

∇ ⋅ u = 0 (1)

∂ u ∂ t + ( u ⋅ ∇ ) u − ν ∇ 2 u = − ∇ p / ρ (2)

For the Reynolds numbers typical for industrial applications, the computational resources required for a Direct Numerical Simulation (DNS) of Equations (1) (2) are exceeding the currently available technical capabilities. The effect of turbulence is normally simplified by solving the Reynolds-Averaged Navier-Stokes (RANS) equations, which are the time averaged approximation of Equations (1) (2). The averaging of fluctuating velocities generates additional terms, known as the Reynolds stresses. To describe these stresses the additional empirical equations, generally in the form of partial differential equations are required to close the computational model. The majority of RANS models are based on the concept of an eddy viscosity, equivalent to the kinematic viscosity of the fluid, which describes the turbulent mixing or the diffusion of momentum. For closure, in this study the turbulence k-ω-SST formulation is used [13] .

∂ k ∂ t + ∂ ( u j k ) ∂ x j = P k − β * k ω + ∂ ∂ x j ( μ + σ k μ t ) ∂ ω ∂ t (3)

∂ ω ∂ t + ∂ ( u j ω ) ∂ x j = γ μ t P k − β ω 2 + ∂ ∂ x j ( μ + σ ω μ t ) + D (4)

where turbulent viscosity is defined as:

μ t = a 1 k max ( a 1 ω , Ω F 2 ) (5)

Far field is assumed at least 100 chord lengths away from the aircraft in x and y direction and +z direction. The -z distance was measured in terms of distance h as it was normal to the ground. A free stream turbulence intensity 0.1% was assumed at the inlet and pressure was discretized to be zero gradient in normal direction at inlet, outlet and the airplane surfaces.

The ground effect aerodynamics was simulated using the steady-state and unsteady solver (U/RANS), closed by the k-ω-SST model for turbulence. Under relaxation is applied for solution of U/RANS equations to increase convergence stability. Second order discretization schemes were used to solve momentum and continuity equations. All scalar variables are solved with the first order accuracy. The residuals for all the equations are allowed to reach a satisfactory convergence of 1/10,000th of the initial values (see Figure 3).

In the case of unsteady simulations at moderately high angle of attack and bank angle, the steady state solution is computed first, which is then used to initialize the unsteady solution to improve the convergence and stability for the unsteady solution.

The simulations were run on a University cluster with 24 processors and the time for convergence for a single point in steady state simulations took an average of 1 - 2 days, while for unsteady simulations it took 4 - 5 days to reach convergence. The average CPU time was about 1.05 min/iteration. Most of the simulations ran from at least 5000 iterations to a maximum of 15,000 iterations.

The results for M = 0.4, Re = 24 million at zero angle of attack is validated against available data obtained at the Netherlands Aerospace Center (NLR)

using their in-house CFD code ENFLOW. The lift coefficient at the above mentioned flight conditions is C_L = 0.188 for our simulations and C_L = 0.197 for ENFLOW simulations. The variation in the lift coefficient between two codes is about 5% and is subject to many differences such as grid and numerical setup.

In close proximity to the ground the airplane wing tip vortices are modified giving a reduced downwash contribution. This leads to increase in the lift force, reduction in the amount of induced drag, onset of the pitching down moment. For illustration purposes, Figure 4 shows a pressure distribution on the CRM surfaces, the ground and in a far field cross-section at flight with altitude h = 0.5 c ¯ , angle of attack α = 8 ∘ and bank angle ϕ = 4 ∘ . Note, the wing closer to the ground has a much weaker tip vortex than a similar vortex on the upper wing, which appears to be slightly elliptical. There are also zones with increased pressure on the ground under the wing and horizontal tail.

Transformations of the wing tip vortices in ground effect produce changes in the aerodynamic forces and moments acting on the aircraft. Figure 5 presents simulated dependencies for the lift, drag and also for the pitching, rolling and yawing moment coefficients. The ground effect increments in the aerodynamic loads increase with increase of the magnitude of the lift and strengthening the wing tip vortices and downwash outside of the ground. For example, at α = 8 ∘ and zero bank angle ϕ = 0 ∘ the increase in the lift coefficient is Δ C L = 0.08 , which is equivalent to increase on 11.2% (see Figure 5, top left plot). Further increase in the lift coefficient takes place at bank angle ϕ = 12 ∘ , Δ C L = 0.217 , which is equivalent to increase on 28.7% (see Figure 5, top right plot). The drag coefficient at bank angle ϕ = 12 ∘ decreases, Δ C D = − 0.013 , this is equivalent to decrease on 11.8% (see Figure 5, top right plot).

The aerodynamic moments are also affected by the bank angle in close proximity to the ground. There is a significant pitching down effect at ϕ = 12 ∘ , i.e. Δ C m = − 0.198 (see Figure 5, bottom left plot). The most important for our objectives are the rolling and yawing moment dependencies on bank angle, shown in Figure 5, bottom right plot. The moment coefficient C_l proportionally

decreases with increase of bank angle ϕ , which is acting in a way as a stiff spring. The yawing moment coefficient increases with increase of bank angle until ϕ = 8 ∘ , but decreases with the change of sign at ϕ = 12 ∘ (see Figure 5, bottom left plot).

The ground effect in aerodynamic characteristics is proportional to the lift force. For CRM configuration we considered high angle of attack runway approach. The airplanes are normally approach landing with deployed leading and trailing edge flaps, which produce a high lift at low angles of attack. Hereafter, we present preliminary set up for a high-lift configuration.

For evaluation of the airplane lateral-directional dynamics in close proximity to the ground the stability-axis lateral-directional equations are considered in the following vector-matrix form (see notations in [16] ):

| r ˙ β ˙ p ˙ ϕ ˙ | = [ N r N β N p N ϕ − 1 Y ¯ β 0 g / V L r L β L p L ϕ 0 0 1 0 ] | r β p ϕ | + [ N δ a N δ r 0 0 L δ a L δ r 0 0 ] | δ a δ r | (6)

The new terms in the state matrix of Equation (6) are N ϕ = C n ϕ ( h ¯ ) ρ V 2 S b 2 I z z and L ϕ = C l ϕ ( h ¯ ) ρ V 2 S b 2 I x x . They represent the rolling and yawing accelerations

induced by bank angle ϕ. The ground effect in this case is equivalent to a kind of “aerodynamic roll stiffness”, which will tend to level the airplane above the runway.

In flight away from the ground, when N ϕ = L ϕ = 0 , the lateral-directional modes are defined by the Roll-Dutch complex-conjugate eigenvalues λ D R = − ξ ± ω n 1 − ξ 2 , the roll subsidence eigenvalue λ R and the spiral mode eigenvalue λ S . It is reasonable to represent the ground effect in the form of a root-loci with a parameter indicating variation of the reduced flight altitude h ¯ = h / c ¯ .

The lateral-directional characteristic equation with account of ground effect can be represented in the following form:

( s − λ S ) ( s − λ R ) ( s 2 + 2 ξ ω n + ω n 2 ) D R − L ϕ ( s 2 + a 1 s + a 0 ) G E = 0 (7)

where

a 1 = N ϕ L ϕ L r − N r − Y ¯ β

a 0 = N β − N ϕ L ϕ L β + ( N r − N ϕ L ϕ L r ) Y ¯ β (8)

Parameter L ϕ varies from zero value in flight with no ground effect ( h ¯ = ∞ ) to its maximum value in close proximity to the ground ( h ¯ = 0.5   -   1.0 ). The N ϕ / L ϕ ratio in expressions a₁ and a₂ (8) has a weak dependence on reduced altitude h ¯ . So, with increase of parameter L ϕ the eigenvalues will move on the complex plane from their initial values λ D R = − ξ ± ω n 1 − ξ 2 , λ R and λ S toward s the values defined by zeros z₁, z₂ of the second order polynomial equation s 2 + a 1 s + a 0 = 0 and one pair of eigenvalues will migrate to infinity. The location of zeros z₁ and z₂ depends on lateral directional coefficients in the expressions for a₁ and a₀ (8). These zeros can be located in the left half of the complex plane, being a complex-conjugate pair, or move to the right unstable half of the complex plane creating an opportunity for onset of oscillatory instability due to ground effect, when a₁ < 0. There is also a possibility for onset of aperiodical instability due to ground effect if a₀ < 0.

A full flight simulation model of a typical transport aircraft has been modified taking into account the aerodynamic dependencies presented in the previous section for the 6-DOF flight simulations in a level trimmed flight in close proximity to the ground. The ground effect is strong in close proximity to the ground and at the altitude exceeding four mean aerodynamic chords practically vanishes.

The eigenvalues of the linearised equations of motion are presented in Figure 8 with variation of parameter h ¯ . The eigenvalues root-loci shows significant transformation of the lateral-directional modes of motion.

The roll subsidence and spiral eigenvalues in close proximity to the ground

are merging creating the oscillatory Roll-Spiral mode with quite significant frequency ω R S = 0.538   rad / s (see Figure 8, bottom plot, α = 8 ∘ ). Along with this change, the Dutch-roll eigenvalues increase frequency from the level of ω D = 1.05   rad / s to ω D = 1.34   rad / s .

There are very little changes in the short-period longitudinal eigenvalues, λ S P and practically no changes in the longitudinal phugoid mode, λ P h . In Table 1 the eigenvalues for the lateral-directional motion modes for flight at h = ∞ and h ¯ = 1 c ¯ , are presented for clarity showing a substantial transformation of the lateral-directional dynamics.

The new factor introduced in the performed eigenvalues analysis was the rolling and yawing moments depending on the airplane bank angle, which was equivalent to the “aerodynamic banking stiffness”. This “aerodynamic stiffness” is strongly affecting the airplane controllability in close proximity to the ground. The airplane responses to aileron and rudder control inputs obtained in the 6-DOF flight simulation are shown in Figure 9 and Figure 10, respectively. The airplane responses to pilot control inputs change in amplitudes and frequencies at low reduced altitude h ¯ = h / c ¯ . This may lead to changes in handling qualities at low altitudes with effect on crosswind landing and onset of pilot induced oscillations.

In crosswind approach-and-landing the aircraft should fly with some nonzero sideslip angle to compensate side-wind. To fly a straight line along the runway the aircraft at the same time should have some non-zero sideslip and bank

angles. Figure 11 shows the required control pilot inputs in trim flight with steady sideslip α = 8 ∘ , ϕ = 4 ∘ and β = 10 ∘ . The control inputs are normalised with respect to maximum deflections. One can see that during landing significant retrimming is required in the longitudinal and lateral control channels and thrust control, and less sensitivity is shown in the directional channel.