Numerics: Viscous Flows, Navier—Stokes Solvers
With the increasing power of computers, it becomes possible to study the aerodynamics of wing-in-ground-effect vehicles by direct numerical modelling on the basis of appropriate viscous flow problem formulations. Similar to computational methods based on Euler equations, this approach enables sufficiently rigorous representation of the geometry of the vehicle. Besides, such a technique is not constrained by technical difficuties related to simultaneous physical modelling by using several similarity criteria (say, Froude number and Reynolds number). It is not restricted either in ranges of variation of the said criteria, so that the future of full scale numerical modelling looks promising.
However, practically, the implementation of numerical approaches introduces certain difficulties. First of all, the numerical treatment of viscous flows for large magnitudes of the Reynolds number is associated with the necessity of solving the Navier-Stokes equations with a very small parameter in front of the higher derivative. This fact gives birth to numerical instabilities due to ill conditioning of corresponding matrices and considerably complicates computations even in two dimensions. The difficulties multiply for the case of unsteady, three-dimensional flow.
An approximate method for computing ground-effect lifting flows with rear separation was proposed by Jacob . The author used an iterative approach combining the three-dimensional lifting surface theory for inviscid incompressible flow with a two-dimensional flow model incorporating compressibility and displacement effects. The author’s analysis implicitly contains an assumption that the aspect ratio of the wing is moderate or large.
Kawamura and Kubo  used a finite-difference method to solve a 3-D incompressible viscous flow problem for a thin rectangular wing with end- plates moving near the ground plane. They employed the standard MAC method (implying the use of the Poisson equation for the pressure and the Navier-Stokes equations) and a third-order upwind scheme. They did not use any turbulence closure models and restricted their calculations to Re — 2000.
Akimoto et al.  applied a finite-volume method to study the aerodynamic characteristics of three foils in a steady two-dimensional viscous flow on the basis of the Navier-Stokes equations with the Baldwin-Lomax turbulence model. To provide modelling of the wake, the position of its centerline was determined by a numerical streamer. The centerline of the wake was represented by a line of segments, extending from the trailing edge of the foil to the boundary of the computational domain. The number of finite volumes used in the calculation was 30×120 in vertical and longitudinal directions, respectively. All calculations were carried out for a Reynolds number equal to 3 x 106. The authors report that the typical CPU time for a set of parameters was of the order of 150 minutes on an alpha-chip workstation.
Steinbach and Jacob  presented some computational data for the airfoils in a steady ground effect at a high Reynolds number. Their approach was based upon an iterative procedure including the potential panel, boundary layer integral method and the rear separation displacement model.
In 1993 Hsiun and Chen solved the steady 2-D incompressible Navier- Stokes equations for laminar flow past an airfoil in the ground effect. Later on , the same authors developed a numerical scheme based on a standard к — є turbulence model, generalized body fixed coordinates, and the finite volume method. They presented some numerical results concerning the influence of Reynolds number, ground clearance, and angle of attack on the aerodynamics of a NACA 4412 airfoil. The range of Reynolds numbers therewith did not exceed 2 x 106.
The Reynolds averaged Navier-Stokes (RANS) approach was also applied by Kim and Shin  to treat a steady two-dimensional flow past different foils, including NACA 6409, NACA 0009, and an S-shaped foil, the latter form providing static stability of longitudinal motion. Transformed momentum transport equations were integrated in time using the Euler implicit method. A third-order upwind-biased scheme was used for convection terms, and diffusion terms were represented by using of a second-order central difference scheme. The pressure field was obtained by solving the pressure Poisson equation. Since a nonstaggered grid was adopted in this method, the fourth – order dissipation term was added in the Poisson equation to avoid oscillation in the pressure field. Two-block H-grid topology was adopted both above and below the foil surface. Two grid points from each block overlapped to ensure flow continuity. For a Reynolds number of 2.37 x 105 adopted for calculations and a 150×120 grid, employed to simulate turbulent flow, 300-500 seconds were required to produce a calculation on Cray C-90 supercomputer.
Hirata and Kodama  performed a viscous flow computation for a rectangular wing with endplates in the ground effect. For this purpose, they used a Navier-Stokes solver, based on a third-order accurate upwing differencing, finite-volume, pseudocompressibility scheme with an algebraic turbulence model to close the system of equations. To be able to treat complex configuration of the flow the, authors used a multiblock grid approach.
Hirata  extended the same technique to attack numerically the problem for a power-augmented ram wing (PARWIG)-in-ground effect. The thrust of the propeller, ensuring power augmentation, was represented by prescribed body-force distributions. However, the Reynolds number for which the calculations were made was somewhat moderate (Re = 2.4 x 105). A similar approach was used by Hirata and Hino  to treat the aerodynamics of a ram wing of finite aspect ratio.
Barber et al.  applied RANS equations with а к — є turbulence model to investigate the influence of a boundary condition on the ground on the resulting calculated aerodynamic characteristics of a foil in two-dimensional viscous flow. The authors aimed at discerning differences in the existing modelling technique of ground effect aerodynamics. Calculated results were presented for Re = 8.2 x 106. In another paper by the same authors , the RANS technique was used to analyze the deformation of an air-water interface, caused by a wing flying above the water surface.
It should be noted that the application of computational fluid dynamics (CFD) at very high Reynolds numbers is not straightforward for both numerical and physical reasons; see Patel  and Larsson et al. . The general problem is that the ratio of the smallest to the largest scales of the flow decreases with an increase in the Reynolds number. Numerically, it means that more grid points are required to obtain a given resolution and physically the nature of turbulence changes, which means that turbulence models developed at low Reynolds numbers might not be valid for high ones. Viscous effects are comparable with inertial ones in the immediate vicinity of the wall. Therefrom, for sufficient resolution, the number of points of the grid in the direction normal to the surface of the body should be much larger than along that surface. As a result, as a consequence of limited computer resources, extremely elongated numerical cells appear near the body surface, causing ill conditioning of corresponding systems of equations and breakdown of most of the solvers; see Larsson . In spite of the progress envisaged in numerical solution of Navier-Stokes equations with the use of large eddy simulation (LES), and ultimately through direct numerical simulation (DNS), the experts do not expect that these methods would be realized earlier than 10 and 20 years, respectively.