On transonic flow models for optimized design and experiment

In the paper the near sonic flow theory for flows with small perturbations to sonic parallel flow is developed. This theory stands on the basis of potential flow of a compressible fluid and enables to receive an exact solution of the flow parameters past transonic cusped airfoils and their geometrical description. Generated airfoil shapes are tested using CFD ANSYS Fluent code to validate the results. Obtained numerical results from all-round commercial code show good accordance with the theory and confirm their value for future work in transonic design.


Introduction
Among various aerospace vehicle speeds, the transonic regime has always posed remarkably difficult challenges to systematic design and experimental techniques.Theoretical methods to understand transonic flow phenomena developed in past time shed light into transonic testing techniques, by solving the underlying physical relations, in various stages of their simplifications.Such analytical models led to the geometrical description of basic airfoil shapes and subsequently to realistic wing sections including qualitative insight into the structure of surrounding transonic flow.At the same time, numerical simulations of these physical relations were developed so that prior to experimental investigations quantitative results were obtained.These, finally, allow for a calibration of windtunnels which are needed to aid the industry in their development of efficient flight vehicles.In this contribution we try to remember some analytically defined methods and airfoil geometries as test cases and compare analytical flow description with numerics, making use of a commercially available computational fluid dynamics code.Our final goal is to arrive at a high degree of understanding typical transonic challenges so that accelerated optimization strategies may be carried out with a reduced set of input parameters, before costly experiments may focus on optimum design cases.

Near Sonic Flow Theory
This chapter was written using [1].The steady state twodimensional potential flow is defined by the continuity equation and irrotationality.
We define velocity potential ‫‬ and stream function ȥ and ‫ׇ‬ the flow angle.
where density as function of Mach number is [2]: This system (3) and ( 4) of nonlinear Cauchy-Riemann equations or so-called Beltrami equations after elimination of ȥ and ‫‬ yields: To avoid the nonlinearity of this basic system we transform the solution to the hodograph plane replacing the physical coordinates x, y with new ones, the flow angle ‫ׇ‬ and Prantl-Mayer turning angle ȣ with a* being the critical velocity.
These new variables lead to define a hodograph plane wherein the basic Beltrami system becomes linear: ) ‫ׇ‬ and ȣ are also functions of a computational working plane obtained from the basic ȣ, ‫ׇ‬ hodograph by conformal (subsonic) or characteristic (supersonic) mapping.For subsonic including sonic conditions conformal mapping defines working plane ȗ.E is the mapping function.
The basic system in ȗ becomes then: ȣ(s,t) is then the real and ‫(ׇ‬s,t) imaginary part of ) ( 1ζ − E .These equations ( 14) and (15) form the linear Beltrami system and elimination of ȥ and ‫‬ leads to a linear Poisson equations: In the supersonic region new characteristic variables occur with a suitable mapping function H. ) The system is then valid in ȗ, Ș plane That is the basic relation to integrate the flow equations for method of characteristics for supersonic flow.This solution allows the integration of physical coordinates x, y by for flows with small perturbations to a sonic parallel flow so that

(
) and by eliminating ȥ and ‫‬ we can obtain a system for physical plane coordinates.The transonic similarity laws containing a similarity parameter ı for reduction of variables for place and state x, y, q, ‫ׇ‬ are: S is positive for supersonic, negative for subsonic and equal to zero for sonic conditions.The basic system (3) and (4) then yields to corresponding Beltrami system for reduced physical plane parameters S and T.  The chamber/thickness parameter is given by: When the cusp is chambered, it is pointing into the flow and it is smoothly passed by the stream so the flow is not forced to change direction around sharp leading edge.The angle of attack is then: The geometry vertex data for family of cambered airfoils is given by: and finally the pressure coefficient: where Ȗ is specific heat ratio.

Numerical Solution
For the numerical simulations the upcoming three variants of cusped airfoils were investigated.The symmetrical "Guderley's cusp", the limit variant with thickness to chord ratio Ĳ = 0.1 and parameter Ȧ/Ĳ = 0.5 and last case with parameters somewhere in the middle of the exact solution bounds, Ĳ = 0.05 and Ȧ/Ĳ = 0.02.
The computational domain was 20x the chord length in all directions.The mesh for all cases was mapped with quad cells with total count of approx.100000 elements, except for the symmetrical case where only the half of the plane could be used.The detail of the mesh for one of the cases is shown on figure 4.1.For flow simulations was used the inviscid -Euler model with ideal gas implemented in a commercial CFD software ANSYS Fluent.The numerical scheme was the AUSM.The only outer boundary condition was set as the pressure far field and it is computed using the gas dynamics equation [4]: Where p 0 is the total or atmospheric pressure and Mach number exactly one for this case.Pressure p then gives the value for the far field.

Results
All three tested cases successfully led to a converged solution using explicit solver after 10 -15 thousand iterations.Contours of Mach number to visualise the simulated flow field are shown on figure 5.1.-5.3.The most important thing that can be noticed from these figures is the fact that no shock appears anywhere near the leading edge or along the airfoils.The velocities at the leading edge are subsonic and along the airfoil the velocity continually rise to supersonic values what forms the obligue shocks propagating from the trailing edge.The flow is smooth for all cases along whole length as the theory suggests.

02111-p.4
The differences between the cases depend on the input thickness and cham parameters.The flow for the symme expected symmetrical with two obligue the trailing edge, but for the thickest and limit airfoil the shock is formed only on the airfoil while the flow is not accelera lower side and it remains shockfree ti velocity near the end of the airfoil i velocity behind.The thin and less ch somewhere between two previous varian shocks but with different strength.thickness and chamber a higher Mach reached.
The way how to compare the theoretical ones is via the pressure described above.Theoretically c coefficient distribution on the airfoil su eq. ( 35) and figures 5. /Ĳ = 0.02.
From correct flow behav contour figures is expectable t distribution along the airfoil s at least tendentiously with the on figures 5.4.-5.6.conf pressure coefficient values lay numerical results almost theoretical.As the values noticeable deviations appear.E separation on the leading edge the trailing edges so the charac with the theoretical assumptio right in these areas.This may b flow speeds start to differ from Mach numbers reached in the 2.3.Inviscid Fluent code also be rotational, what the analytic

Conclusions
Theory of two-dimensional t airfoils was developed on t potential flow and it is used f generation.The numerical exp data on parameters and flow s Obtained results confirmed w numerical and exact analytic sonic flow theory.More than a this work is aimed to show t already developed theories fo designs that can be initial case real prototypes.These cases easily comparable with resu CFD codes.These can be con reliable tool for future steps and optimization.

Acknowledgement
This work has been supported Czech Technical University SGS13/180/OHK2/3T/12 vior already obvious from that the pressure coefficient urface will also correspond e theoretical lines and data firm that.In areas where y between 0.3 and -0.8 are identical with dashed rise from these bounds Even when there is no flow e and shocks appear only at cter of the flow corresponds ons some differences appear be given by the fact that the m a near sonic flow here as ese areas vary from 0.6 to probably allows the flow to cal potential flow neglect.
transonic flow past cusped the basis of compressible for optimized profile shape periment enables to receive tructures past these airfoils.well the accordance between cal data by means of near any concrete practical value the possibility of using the or creating first systematic es for optimized shapes for s in transonic regime are ults obtained from modern nsidered as useful, fast and in transonic airfoils design zky, Proc.Int.Symp on neering mechanics, Nagano Exact SolutionSolving previous hodograph relations allows to derive the formulae defining the shape, flow conditions and pressure coefficient for cusped airfoils in a uniform sonic flow 1 = ∞ M[3].The schematic wiev on a cusp with all defined parameters is on figure 3.1.

Figure 3 . 1 .
Figure 3.1.The cusp parameters.Thickness to chord ratio is Ĳ and chamber to chord ratio Ȧ.The solution is exact for 0 → τ and practically valid for slender airfoils with 5 .0 ≤ τ .The case with chamber to chord ratio 0 = ω is the symmetrical one also known as "Guderley's cusp".Limit of validity for chambered airfoils is given by ratio 5 .0 / ≤ τ ω