3D Cascade Analysis On Garabedian-Korn Supercritical Profile – Aerospace Engineering

Supercritical Airfoils

In the recent years, the computational fluid dynamics field has made huge strides in expanding the scientific research. The purpose of this paper is to give a clear analysis and design of the 3D cascade analysis on the blades made using the Garabedian-Korn supercritical profile. The paper is written from a designer’s perspective with the aim of reviewing the analytical and computational schemes utilized to make relevant decisions. The aircrafts seek to cruise at faster speed with lower consumption of fuel, improved ecosystems to avoid air pollution, and most essentially to improve the design to ensure all the above is attained. Before any design is implemented a prototype is tested on land. The 3D cascade analysis is a concept reviewed under computational fluid dynamics for turbomachinery in aerospace engineering. Supercritical airfoils are a class of transonic airfoils.  They are said to operate with subsonic inlet. The inlet has exit flow velocities and with entrenched regions of supersonic flow adjacent to the airfoil surface. The Garabedian-Korn supercritical profile refers to the presence of velocities in the flow field above the critical speed. This speed is also referred to as the sonic speed. Garabedian-Korn airfoil is a supercritical profile that has been widely used as a benchmark to authenticate numerical routines for CFD. G-K airfoil was established in the early 1970’s using the hodograph method.

There are 11 passages in the transonic turbine blade when analyzed on a linear scale. A flow field is obtained when the end wall turbulent boundary layer is allowed to develop in a long inlet section upstream of the cascade (Chopra, 2007). The Garabedian-Korn airfoil was established in the early 70s using the hodograph method. There was a slight drawback in that the coordinates of the airfoil contained a discontinuity at the sonic line. Vassberg & Jameson presented a study entitled “asymptotic convergence with extreme mesh refinement for inviscid subsonic and transonic flows of several widely used computational fluid dynamics methods”. The study was conducted to test if the proposed shock free airfoils are truly shock free (Jameson, 2011). At first asymptotic behavior of transonic flow past the G-K shock free airfoil was examined. The first concern raised was if the shock free airfoil was truly shock free? The G-K airfoil was designed to be shock free at Mach 0.75.

It is not considered shock free when the mesh is adequately refined. It is shock free at a little higher Mach of 0.751. The current aero engine design involves the use of supercritical profiles for high performance compressor stages and hence G-K profile is well suited for modern compressor stages. Feng Liu and Antony Jameson applied multi-grid Euler method for cascade flow calculations. He used explicit finite volume method for Euler equation; this method is capable of capturing qualitative features of secondary flow due to inlet side wall boundary layer. Weber Anton investigated 3D transonic flow in a compressor cascade; he conducted an experiment and numerical study of the transonic flow through a linear compressor cascade with end walls. (Jameson, 2011). This airfoil is researched to determine the design attributes as well as the relation with its neighborhood. Several flow solutions are provided. They establish a grid converged force and moment values based purely on the sequence of meshes. In aerospace engineering, there are 2D and 3D axial designs for the flow turbines and the axial flow compressors (Bauer, 1977). The innovation is categories under turbomachinery aerodynamics. The research study by Collyer states that,

Transonic Turbine Blade Analysis

“It is annulus in nature that they flow through the annulus space available to the turbines and is often not exactly uniform. Accurate prediction of turbine cascade flow and pressure losses is a challenging task to aerospace engineers and designers.”

The Garabedian-Korn airfoil is about 11.6% thick and has almost 2% camber, and contains a cusped TE since . The transonic flow is a serious point of study in the aircraft design as it is the most competent regime for long series transport aircraft. The studies show very accurate values for the slope and arc-length. The demerit of the design is fixed by reconstruction of new coordinates of the airfoil using a conformal mapping. The software used to show the design specifications shows the reconstructed contours. The contours are now non-dimensional basing the argument on the maximum length of the chord line. It is rotated to place the true leading edge point at the origin and the trailing edge point. The design of the axial flow compressors, most specifically the rotors requires one to comprehend the flow of the blades. In the aero dynamical machinery the air flows and turns to three dimensions. At this state one need to know how to handle it. This is done with the help of computational fluid dynamics. In most nations, the military is mandated with the task of coming up with new designs and improving the existing ones.

As shown in the equations below, several characteristics are necessary as they act as the parameters to adjust for a given result to be obtained. The cascade idea is a common one and it is referred to as a stationary arrangement. It is a modification that makes the system more reliable. In such a design the airfoil is subject to flow either linearly or as a two dimension. In the rotor, the rotating airfoil in a blade moves from the root of the blade to the tip of the blade while it rotates. This poses a great challenge to the designers. Both the rotor and the stator in their different motions are in space and they do not possess a flat arrangement. During the rotation, the aerodynamics creates a three-dimensional flow. In the current market, the compressor designers opt to go for the 3D blades so as to keep the aerodynamics of the flow nearer to the two-dimensional theory which is commonly understood. The design of the blades is described as a twisted. The twist is required to improve the design and are essential in creating a 3D airfoil. The range of an aircraft is well defined by the Breguet equation,

Research Study on Garabedian-Korn Airfoil

Where,

V	Cruising speed
Sfc	Specific fuel consumption of the engines
L/D	Lift drag ratio
Wf	Weight of the fuel burnt
Wo	Final weight at the end of the flight

The drag coefficient in a subsonic flow is obtained from the classical aerodynamic theory as,

CDo	Zero lift drag
CL	Lift coefficient
AR	Wing aspect ratio

Where,

The L/D is maximized when the aircraft is flow at a lift coefficient such that the two terms are equal,

It is plausible to conclude that one flies faster when the L/D is at its maximum (Uchida, 1954).

To calculate for the compressible and transonic flow past a single aero foil with a grant for viscous effect, a numerical computation method is implemented. The fluid flow determines the turbulence management levels of a given gadget when cruising in the atmosphere. The design specifications need to be made more soberly. It provides that the boundary layer is fully attached over the aero foil surface. The method has been developed by combining an iterative scheme for the inviscid flow, originally established by Garabedian-Korn, with an integral method for calculating the compressible turbulent boundary layers. The inviscid scheme is modified to incorporate a boundary condition on the aero foil surface which is imposed on the velocity normal to the surface, with a corresponding boundary condition for the blade. The flow analysis is done using the ANSYS workbench with CFX or Fluent as a solver. Garabedian-Korn settled these computer codes to study the super critical wind. As a result, they managed to reduce the boundary layer separation by shifting shock waves towards the trailing edge of the wind and making them as weak as possible. The team used complex attributes in hodograph to design shock less airfoil that generates smooth transonic flow. The designation profile from the two researchers, Garabedian and Korn, is commonly referred to as “75-06-12”.

The compressor cascade is well demonstrated as a concept using the compressor blades. Within the compressor blades, the flow moves from a low static pressure at inlet towards a higher static pressure at exit. The fundamental difficulty in compressors is getting the flow to negotiate this pressure rise without generating high loss or getting separated. The axial compressor designer opts to choose an appropriate level of blade loading (Bauer, 1977). There have been several improvements on the work done by Garabedian and Korn. A computer program named VGK was developed based on the same work by Garabedian-Korn studies (Smith & Calvert, 1976). In their studies, the viscous effects are included by an amalgamation of a scheme for inviscid flow calculations which determines a pressure distribution on the aero foil surface and along the blades (Clapworthy & Mangler, 1974). An integral routine for predicting the development of turbulent boundary layer in a given pressure distribution providing the boundary layer is used. More research studies have been developed to try and bridge the gap and overcome the blade limitations (Collyer, 1978). In the present development of the problem of including viscous effects of the boundary layer method employed by Firmin and Jones is retained but the inviscid scheme is replaced by a more accurate one than that proposed by Garabedian-Korn. The scheme comprises of the conformal transformation of the aero foil into the interior as developed by Sells (Collyer, 1978). The solution is combined with a recurrent solution of the actual irrotational, isentropic equations in a manner similar to that used by Murman and Cole. The procedure seeks to employ the consistent reiteration of the inviscid scheme’s current pressure on the aero foil surface. The calculations pertaining the scheme are done and solutions show an improvement in the interaction with boundary layer and other aerodynamic attributes (Collyer, 1978). The wind tunnel is a significant tool that describes what takes place but it hardly provides guidance on why a particular event occurs. The research teams at NASA and other aircraft bodies work effortlessly to improve the airfoil performance beyond the current designs.

Axial Flow Compressor Design

This research paper seeks to discuss the aero dynamical gap  discover the limitations of the existing system as well as come up with mathematical solutions that can improve the design by cascading the flow in a 3D system. From the studies done by other researchers, one clearly gets an idea of the angle of attack and Mach (Davis, et al., 1965) at which the flow analysis is conducted. There is need to use the same results to find out what would happen in a cascade condition. The creation of a cascade profile for Garabedian Korn superficial profile is a good place to start. The cascade profile creation depends on the blade angle, twist angle, and tip clearance angle (Hawthorne & Galapatti, 1976). The research data analysis is carried out and the effectiveness of the ANSYS workbench and its solver CFX and Fluent are analyzed.

The meshing of the body should be very fine as there is need to gather data of the analysis and check the intensity of the shock produced which is truly negligible or it can be termed as shock-free. The rotation speed and direction of rotation of the blades is another key consideration as the research gap. The blades being discussed are those of the rotor in a compressor. The flow conditions for the analysis is not specified which tends to create another gap. Research carries on about a similar kind of blades of different compressor and an assumption is made for analysis.

References

Bauer, P. F., 1977. Supercritical Wing sections III. New York: Springer Science & Business Media.

Borland, C. J., “XTRAN3S – Transonic Steady and Unsteady Aerodynamics for Aeroelastic Applications,”AFWAL-TR-85-3214, Air Force Wright Aeronautical Laboratories, Wright-Patterson AFB, OH, January, 1986

Chopra, A., 2007. Elementary Cascade Theory and Gas Turbine Performance. Kanpur: Indoor German Winter Academy.

Clapworthy, G. J. & Mangler, K. W., 1974. The behaviour of a conical vortex sheet on a slender wing near the leading edge, London: ARC/R&M-3790.

Collyer, M., 1978. An Extension to the Method of Garabedian and Korn for the Calculation of Transonic Flow past an Aerofoil to Include the Effects of a Boundary Layer and Wake. London: Her Majesty’s Stationery Office.

Das, h. S. a. A., 1996. Recent Research on Cascade-Flow Problems. ASME, 88(1)(1), pp. 8-9.

Davis, B. M., Berry, C. J. & Rogers, E. W., 1965. An experimental investigation of the interaction between a forward-facing step and a laminar boundary layer in supersonic, low-density flow, London: ARC/R&M-3506.

Drela, M. “Newton Solution of Coupled Viscous/Inviscid Multielement Airfoil Flows,”, AIAA paper 90-1470, presented at the AIAA 21st Fluid Dynamics, Plasma Dynamics and Lasers Conference, Seattle Washington, June 1990.

Garabedian, P. and McFadden, G., “Computational Fluid Dynamics of Airfoils and Wings”, Proc. of Symposium on Transonic, Shock, and Multi-dimensional Flows, Madison, 1981, Meyer, R., ed., Academic Press, New York, 1982, pp. 1-16.

Hawthorne, R. W. & Galapatti, R., 1976. Cascades with arbitrary end wall contraction, Northwington: ARC/R&M-3813.

Harlow, F. H. (2004). “Fluid dynamics in Group T-3 Los Alamos National Laboratory: (LA-UR-03-3852)”. Journal of Computational Physics. Elsevier. 195 (2): 414–433. Bibcode:2004JCoPh.195..414H. Doi:10.1016/j.jcp.2003.09.031.

Hunt (1998). “Lewis Fry Richardson and his contributions to mathematics, meteorology, and models of conflict”. Annual Review of Fluid Mechanics. 30. Bibcode:1998AnRFM..30D.13H. Doi:10.1146/annurev.fluid.30.1.0. “The Legacy of Group T-3”. Retrieved March 13, 2013.

Jameson, J. c. V. A., 2011. Further Studies of Mesh Refinement: Are shock-free airfoils truly shock free?. 20th AIAA Computational Fluid Dynamics, 1(1), pp. 23-35.

Jameson, Antony, “Acceleration of Transonic Potential Flow Calculations on Arbitrary Meshes by the Multiple Grid Method”, AIAA Paper 79-1458, Fourth AIAA Computational Fluid Dynamics Conference,Williamsburg, July 1979.

Murman, E.M. and Cole, J.D., “Calculation of Plane Steady Transonic Flows”, AIAA Journal , Vol 9, No 1, pp 114-121, Jan 1971. Reprinted in AIAA Journal, Vol 41, No 7A, pp 301-308, July 2003

Smith, J. L. & Calvert, J. W., 1976. A digital computer program for the subsonic flow past turbomachine blades using a matrix method, 1976: ARC/R&M-3838.

Uchida, S., 1954. Calculation of Compressible Cascade Flow by the Method of Flux Analysis. Journal of the Aeronautical Science, 21(4), pp. 237-250.

Milne-Thomson, L.M. (1973). Theoretical Aerodynamics. Dover Publications. ISBN 0-486-61980-X.

Richardson, L. F.; Chapman, S. (1965). Weather prediction by numerical process. Dover Publications.