Numerical solution of inviscid and viscous laminar and turbulent flow around the airfoil

This work deals with the 2D numerical solution of inviscid compressible flow and viscous compressible laminar and turbulent flow around the profile. In a case of turbulent flow algebraic BaldwinLomax model is used and compared with Wilcox k-omega model. Calculations are done for NACA 0012 and RAE 2822 airfoil profile for the different angles of upstream flow. Numerical results are compared and discussed with experimental data.


Introduction
Mathematical models used in this work are based on the solution of the inviscid compressible flow and viscous compressible laminar and turbulent flow around the airfoil.
Computational domain represents NACA 0012 and RAE 2822 airfoil profiles.The results show differences between the uses of each numerical model and experimental results.

Governing equations
The two-dimensional viscous compressible flow is prescribed by the system of Navier-Stokes equations in following conservation form This system is closed by following equation In the equations above ȡ denotes density, (u, v) are components of local velocity in x and y direction respectively, e denotes total energy per unit volume, p is pressure given by previous equation of state, Ĳ ij is shear stress, q i represents heat flux, ȝ represents dynamical viscosity.
Following constants were used in previous equations: ț is isentropic exponent equalled to 1.4, Pr is Prandtl number equalled to 0.7.
Viscous compressible turbulent flow represents system of Reynolds averaged Navier-Stokes equations (RANS) which is formally the same as (1) with enclosed used model of turbulence which is algebraic Baldwin-Lomax or two-Wilcox k-Ȧ model [1] in this work.
Inviscid flow is represented by the system of Euler equations which is simply reached after neglecting of viscosity in viscous fluxes on the right hand side in the system (1).
where vorticity is in 2D case defined as van Driest function given as follow and friction velocity by formula Turbulent viscosity in outer region is given by Function F w is determined by the relation where F w is the maximum of the function bellow and y max is the distance from the wall in which F(y max ) = F max holds and In Baldwin-Lomax turbulent model following constants are used: ț = 0.4,A = 26, Į = 0.0168, C cp = 1.6 and C KL = 0.3.

Wilcox k-Ȧ model
The two-equation model is given by transport equations for two characteristic scales of turbulent motion, in this case turbulent energy k and specific dissipation rate Ȧ.

Numerical method 4.1 Numerical scheme
Finite volume method was applied on cell centred nonorthogonal structured grid with quadrilateral cells.For numerical solutions Lax-Wendroff scheme in McCormack modification was used with predictor step and corrector step as follow ( ) The Jameson's artificial dissipation model was applied to damp the oscillations.

( )
) Solid wall condition was realized by the adding of virtual cells.In a case of inviscid flow velocity components were prescribed so that sum of velocity vectors equals zero in its tangential component.In a case of viscous flow velocity components were prescribed so that sum of velocity vectors equals zero on the wall.

Numerical results and discussion
Mach number isolines are shown in all pictures together with inlet flow parameters below where Ma corresponds to the inlet Mach number and Į is attack angle of inlet flow.Numerical results were validated for different setting of artificial dissipation to damp the oscillations and in a case of viscous model the influence on the physical viscosity was checked.
The set of the following six figures corresponds with results of the calculation of transonic inviscid flow for NACA 0012 and RAE 2822 airfoil respectively.This configuration was set as a reference for the testing of behaviour of the inviscid model with regarding to the proper setting of the artificial dissipation.Mach number isolines and pressure coefficient C p are compared with [2].The results of turbulent flow around the NACA 0012 profile are shown also at the figures above.In this case inlet Mach number was set to Ma = 0.15 and angle of attack Į = 10.0º,Reynolds number Re = 2.88×10 6 .Pressure coefficient is compared with experimental data by [3] for the upper surface with free transition.The deviation close the leading edge is visible and needs to be more investigated.It can be caused e.g. by coarse grid spacing in x axis around the leading edge.Also the problem of artificial dissipation model needs to be solved in the next study to reduce the influence of additional numerical viscosity that could be also the source of incorrect behaviour.

Conclusions
The developed software was successfully tested for some basic types of the calculations of inviscid and viscous turbulent flow around the NACA 0012 and RAE 2822 airfoil.Deeper testing still needs to be done in a case of viscous turbulent flow to verify the possibilities of different numerical approaches as well as the testing of another modification of two-equation k-Ȧ model.Also the deeper investigation will be done at a field of the transition between the laminar and turbulent flow.
Inlet boundary conditions were realized for inviscid flow as follow: inlet velocity Ma together with angle of attack Į, density ȡ and total energy per volume e were set; pressure p was extrapolated from the flow field.In a case of viscous flow the inlet pressure p was set.Outlet boundary conditions were the same for the both type of flow.Outlet pressure was set and other variables extrapolated from the flow field.