"Blade Film Cooling by Underexpanded Transonic Jet Layers"

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1 172 PAPER 5 Gehrer, A., Woisetschläger, J., Jericha, H. "Blade Film Cooling by Underexpanded Transonic Jet Layers" ASME Paper 97-GT-246

2 173 ABSTRACT The evolution of increasing turbine inlet temperature has led to the necessity of full-coverage film cooling for the first turbine vane and blade. A new approach using high speed wall jets for blade cooling has been proposed by the authors. In this paper a 2-D upwind-biased Navier-Stokes code is used to calculate the aerodynamic behaviour of these surface coolant jets in a linear cascade. Special emphasis is put on the investigation of the coolant flow around the leading edge. Various numerical results concerning the leading edge flow are presented in detail, showing a strong tendency of these jets to bend towards convex surfaces. Finally, Schlieren pictures taken at the Institute s transonic cascade on a 140mm chord length blade are presented for comparison. NOMENCLATURE Symbols: A + =26 l L M P P * T T 0 u,v,w x y y + δ κ constant for the turbulence model mixing length chord length Mach number total pressure coolant/total pressure main flow critical pressure coolant/total pressure main flow static temperature total temperature cartesian velocity components chordwise coordinate minimum dis tance to any wall nondimensiomal distance from the wall boundary-layer thickness ratio of specific heats ω vorticity µ t eddy viscosity µ 0 laminar viscosity at stagnation Indexes: c coolant m main flow 1 at inflow boundary INTRODUCTION The objective of this work is to investigate underexpanded jets for a novel turbine blade film cooling system in a linear cascade, especially their aerodynamic behaviour around the leading edge. Since these underexpanded jets have a strong tendency to bend towards the surface a higher cooling efficiency is expected due to the improved cooling film attachment. The effect of air flows bending around curved surfaces is generally connected with the name "Coanda-effect" and originally describes subsonic flows attaching to the surface by their turbulent structure (e.g. Fernholz and Wille, 1971; Ameria and Dybbs, 1993). Several experiments with underexpanded jets bending towards a convex surface have been performed and published (Gregory-Smith and Gilchrist, 1987; Gilchrist and Gregory-Smith, 1988; Gregory-Smith and Hawkins, 1991; Gregory-Smith and Senoir, 1994). These curved supersonic jets have a complex structure with compression and expansion waves in the underexpanded core of the jet, with an outer free shear layer of the jet and a boundary layer towards the surface. Since these underexpanded jets bend towards convex surfaces it has been suggested (Jericha et al., 1995) that its use might be considered for high temperature turbine blade film cooling. Cooling slits are suggested instead of holes in order to preserve this major property and also to cover a large area of the blade.

3 174 A preliminary investigation was performed with an acrylic glass model of a cooling slit (Woisetschläger et. al. 1995) which then provided the basis for a cooling slit design inside the linear blade cascade at the Institute of Thermal Turbomachinery and Machine Dynamics, TU Graz (s. fig. 1). CO 2 window section cooler compressor station Fig. 1: Linear cascade, test section The test section was designed to encompass a combined cycle proposed by Jericha et al. (1995) using high pressure steam as cooling medium for the high temperature gas turbine. In this case there would be no need of additional blowing energy. Since it is not possible to run the linear cascade test rig with the same flow temperatures as a real turbine an aerodynamically similar test section was designed using CO 2 as cooling flow medium. A detailed description of the measurement techniques has been given by Woisetschläger et al. (1996). In the work presented here a 2-D Navier-Stokes code, based on the unsteady Euler code of Sanz et al. (1995), is used to demonstrate the effects described above and to investigate the influence of the supersonic cooling flow on the main flow field. For comparison with experimental data the blade profile and the slit geometry of the test section and the respective pressure ratios and temperatures have been investigated. Since this Navier-Stokes code is limited to single-phase flow problems, the CO 2-flow in the cooling slit is calculated separately from the main flow. The respective results for velocity, pressure and density in the throat of the slit, where the average Mach number becomes unity, are then used as inflow boundary conditions for the main flow. NUMERICAL METHOD The unsteady, Reynolds averaged Navier Stokes equations are treated in conservative form and discretized in space using a cell-centred finite volume formulation with a quadrilateral structured cell system and in time using the Euler implicit method. air Inviscid Fluxes On each cell boundary the inviscid fluxes are evaluated by an upwind scheme based on the approximate Riemann solver of Roe (1981). To raise the order of spatial accuracy the inviscid fluxes are evaluated using a 3rd order accurate TVD scheme. The TVD scheme is based on the MUSCL type of an upwind scheme which consists of a projection stage and an evolution stage. In the projection stage, left and right states at each cell interface are determined by extrapolating the cell-centred values of the conservative variables toward the cell interface according to the third-order upwind-biased scheme with a differentiable limiter (Anderson et al., 1985). In the evolution stage the inviscid flux is evaluated by solving the Riemann problem between left and right states using Roe's approximate Riemann solver. The implicit portions of the inviscid flux vectors are simplified by defining an approximation to the true linearization. Therefore, only a first order accurate scheme is considered for the implicit side whereas the full high-accuracy scheme on the explicit side is used (Sanz et al. 1995). Viscous Fluxes In order to construct the numerical viscous flux vector at the cell interfaces it is necessary to evaluate first-order derivatives of the velocity components and the sound speed, which is done in a centraldifferencing manner, using Green's theorem (e.g. Furukawa et al. 1991). The time linearization of the viscous flux vector is performed by applying the thin-layer approximation for the implicit side of the equations. Turbulence Model A two-layer algebraic model based on the mixing length concept is used for turbulence closure. In the near wall region the mixing length is computed using the Prandtl-Van Driest formula while in the outer region and in the wake it is kept constant to a fixed fraction of the shear layer thickness δ, according to the standard relation (Kwon et al, 1988). l outer = δ In the present work an algebraic criterion, which resembles the features of the Baldwin Lomax (1978) model, but which implicitly introduces a cut-off criterion for the vorticity field based on the distance from the wall, is used to estimate the boundary layer thickness (Arnone et al. 1996). If y denotes the distance normal to the wall, the value y max at which the function y + y 1 + G( y ) = y e A dy y ω 1 0 reaches its maximum is assumed as turbulent length scale. The boundary layer thickness is then obtained from the relationship: δ= y max

4 175 Boundary Conditions In cascade-like configurations, there are four different types of boundaries: inlet, outlet, solid wall and periodicity. In the present cell-centred scheme, phantom cells are used to handle all boundaries. According to the theory of characteristics, flow angle, total pressure, total temperature and isentropic relations are used at the subsonic axial inlet, whereas all variables are prescribed at the supersonic inlet. At the subsonic axial outlet the average value of the static pressure is prescribed, density and velocity components are extrapolated, whereas all variables are extrapolated at supersonic axial outlet. On solid walls, the pressure is extrapolated from the interior points and the no-slip adiabatic condition is used to compute density and total energy. The periodicity is imposed by setting periodic phantom cell values. GEOMETRY, GRID SYSTEM The blade profile with its cooling slits as it is built in the test section and the main design parameters are given in fig. 2. All calculations presented in this paper concentrate on the rather difficult situation near cooling slit #1, slits #2 and #3 are not subject of these investigations. Fig. 3.a: Periodic C-type grid for the main flow without cooling jets (251 x 30 nodes) blade chord length 143 mm blade pitch 98.6 mm cooling slit throat 0.2 mm design inlet angle * 55 design outlet angle * 28 * measured from circumferential direction cooling slit # cooling slit # Fig. 3.b: Periodic C-type grid for the main flow with cooling jets (274 x 40 nodes) cooling slit #1 x chord length L = 142mm Fig. 2: Geometry of the blade and cooling slits An algebraic mesh generator, developed by Gehrer et al. (1996), based on Bézier curves and Bézier surfaces was applied to generate a periodic C-type grid for the main flow field, which is shown in fig. 3. Two different grids were used to calculate the linear cascade flow. Fig. 3.a shows the grid for the calculations without coolant which consists of 251 x 30 nodes. A refined version of this grid (s. figs. 3.b, 3.c), consisting of 274 x 40 nodes and including the throat of the cooling slit, is used to calculate the main flow with coolant. Fig. 3.c: Area around cooling slit Fig. 3.d: Mesh for the flow inside the cooling slit (30x27 nodes)

5 176 Fig. 3.c shows the area around the cooling slit of the mesh in fig 3.b and finally fig. 3.d displays the mesh for the flow field inside the cooling slit (30 x 27 nodes). RESULTS AND DISCUSSION Three cases have been investigated. The respective boundary conditions, which were derived from experimental data, are given in table 1: TABLE 1: Boundary conditions case1 case2 case3 no coolant P=2 P * =1.088 P=3 P * =1.632 main inlet flow angle * flow (air) total pressure [bar] total temperature 40 C 40 C 40 C outlet static pressure [bar] coolant inlet flow angle ** (CO2) total pressure total temperature - 20 C 20 C outlet supersonic outlet jets shows reasonable agreement with experimental data. Furthermore, streamlines in the area of the leading edge (fig. 4.b), density contours (fig. 4.c) and pressure contours (fig. 4.d) illustrate the flow situation for case1. M-isentropic Experiment no filmcooling Fig. 4.a: Case 1(no cooling): isentropic Mach number distribution comparison with experiments x/l * measured from circumferential direction ** measured from the middle axis of the cooling slit Numerical results, concerning mass flow, Reynolds numbers and exit flow angles are shown in table 2: TABLE 2: Numerical Results case1 case2 case3 Fig. 4.b: Leading edge: streamlines for case 1 no P=2 P=3 coolant main inlet mass flow[kg/s/m] flow outlet mass flow[kg/s/m] flow angle * Reynolds number 1.65E E E+06 based on chord coolant mass flow[kg/s/m] blowing ratio =(coolant mass flow /inlet mass flow) 1.32 % 2.04 % Reynolds number based on throat * measured from circumferential direction 9.37E E+04 Results for the main flow without cooling jet (case1) Fig. 4.c: Density contours for case 1 In order to validate the numerical results, in fig. 4.a, a comparison of calculation with experiment is presented. The isentropic blade surface Mach number distribution for the main flow without cooling

6 177 even in the region of the highly curved leading edge radius and even in a direction opposite to the incoming flow. The numerical results given in table 2 show a slight reduction of both inlet mass flow and exit flow angle, which indicates that the high speed cooling jet enhances the overall blade vortex. Fig. 4.d: Pressure contours for case 1 Results for the flow inside the cooling slit (e.g. case2) In fig. 5 computational results for the CO 2 - flow inside the cooling slit (e.g. case 2) are presented. Fig. 5.a: Cooling slit: Mach number contours Fig. 6.a: Density contours for case 2 u[m/s] u (NS-calculation) M= y[m] Fig. 5.b: Cooling slit: Velocity distribution across throat In fig. 5.a the Mach number contours illustrate the growth of the boundary layer inside the slit, which is, due to the relatively low Reynolds number (s. table 2) of about 10 4, rather pronounced. In fig. 5.b, the velocity distribution across the throat of the slit is displayed. As mentioned above, the calculated velocity, density and pressure distributions across the throat (see also fig. 3.c) are prescribed as supersonic inflow boundary for the main flow for cases 2 and 3. However, the result for the supersonic part (throat to outlet in fig. 3.d) is not used for the main flow calculation. It should be noted that also the subsonic boundary layer region in the throat was treated as supersonic inflow boundary for the main flow. Results for the main flow with cooling jet (cases 2 and 3) Regarding the flow with coolant, the respective density contours in figs. 6.a and 6.b clearly demonstrate that it appears possible to keep the cooling layer close to the surface of the blade, Fig. 6.b: Density contours for case 3 A more detailed view of the domain of interest in fig. 8 illustrates that the outflow of supersonic cooling medium from the slit forms a cooling layer around the leading edge of the blade. Especially the Mach number distributions in figs. 8.e and 8.f show that the jet expands to transonic velocities and, at the same time, due to the effect of continuous expansion, the film adheres very closely to the surface. This expansion can also be noticed as a pressure drop, which is clearly visible in the pressure contours.(s. figs. 8.a and 8.b). For purpose of comparison, the Schlieren visualisation of the leading edge flow is presented in fig.7 which can be regarded as a qualitative verification of this effect.

7 178 suction side blade pressure side incoming flow direction Fig. 7: Schlieren visualisation of the leading edge flow: main flow with the underexpanded cooling film at a pressure ratio of P = 2 circumferential direction Fig. 8.e: Mach number distribution, case 2: M max = (in the core of the jet). The gridlines, indicated by the numbers 1,2,3 and 4 have been used to evaluate the velocity profiles given in fig. 9.b. Regarding the turbulent viscosity values in figs. 8.g, 8.h, it is obvious, that the outer free shear layer of the jet causes very high turbulence intensities which, especially in the region of opposite flow directions, are the main reason for rather high losses of kinetic energy in the underexpanded core of the jet. The streamlines in figs. 8.c and 8.d show a distinct vortex in this area. circumferential direction Fig. 8.f: Mach number distribution, case 3 M max = (in the core of the jet) Fig. 8.a: Pressure contours, P=2 Fig. 8.b: Pressure contours, P=3 incremental step size: 0.01 bar Fig. 8.c: Streamlines, case 2 Fig. 8.d: Streamlines, case 3 Fig. 8.g: Eddy viscosity Fig. 8.h: Eddy viscosity (µ t /µ 0) contours, P=2 (µ t /µ 0) contours, P=3 (µ t /µ 0) max=1013 (µ t /µ 0) max=1670

8 179 Mis no filmcooling P=2 P= X/L 2 Suction Side 1 Pressure Side Fig. 9.a: Isentropic Mach number distribution in the area near the leading edge The isentropic Mach number distribution in the area around the leading edge (s. fig. 9.a) shows a significant peak shortly after the coolant is injected at the pressure side of the blade. This means that the jets definitely accelerates the main flow (in flow direction of the jet) around the leading edge, which is also clearly visible in fig. 9.b, where the respective velocity profiles are given. The locations of gridlines used to evaluate these profiles are displayed in figs. 8.e and 9.a. T/T no filmcooling P=2 P=3 Pressure Side Suction Side tangential velocity [m/s], scale: 1 increment = 50 m/s y/l (normal to the wall) P = 3 no cooling P = 2 Fig. 9.b: Velocity profiles in the area of the leading edge. The respective locations are denoted by the numbers These positions are also shown in figs. 8.e and 9.a Fig. 10: Predicted temperature at the wall, normalised with the main flow total inlet temperature Finally, the predicted temperature distribution at the (adiabatic) wall, normalised by the main flow total inlet temperature is presented (s. fig. 10). It should be kept in mind, that the results from the CO 2 flow calculations (total temperature, CO 2 : K) inside the cooling slit were transferred to the main flow field (air), keeping the same pressure, velocity and density, which results in a total temperature of about 193 K. Under these conditions, the calculations predict, that the cold cooling jet fluid remains along the whole suction side, its temperature rises due to friction, heat transfer and mixing with the main flow. Finally, it can be noticed, t hat all numerical results for case 3 are qualitatively similar to the results for case 2, which means that this range of pressure ratio P seems possible for a strong adherence between this type of cooling film and the blade surface. Investigations, presented by Gilchrist and Gregory-Smith (1988) and Woisetschläger et al. (1995), show that further increase of pressure ratio P leads to a low speed recirculation zone between jet and surface in the boundary layer which then causes a jet "breakaway" from the surface. CONCLUSION Underexpanded cooling films have a strong tendency to bend towards the surface which makes them interesting especially for cooling of the leading edge area. In this work, 2D Navier Stokes calculations were used to demonstrate, that transonic pressure ratios X/L

9 180 of P = 2 and 3 can provide a sufficient cooling layer around the leading edge area. The respective computational results showed that these transonic wall jets adhere very closely to the surface, even in regions of opposite flow directions. Ongoing numerical and experimental investigations will deal with velocity, turbulence and density changes due to underexpanded surface coolant films with and without upstream flow distortions. Furthermore these investigations are expected to provide data concerning the heat transfer and temperature situation. These data then will be used to evaluate the whole cooled gas turbine or even the whole cycle thermodynamically in order to give a full answer concerning the amount of additional energy for blowing and aerodynamical losses compared to conventional film cooling. ACKNOWLEDGEMENT The authors gratefully acknowledge the support of the Austrian Science Foundation (FWF) supporting this research as well as ongoing research in efficiency improvement of thermal energy production. Gregory-Smith, D.G. and Senoir P., 1994, " The effects of base steps and axisymmetry on supersonic jets over coanda surfaces", International Journal Heat and Fluid Flow, Vol.15, pp Jericha, H., Sanz, W., Woisetschläger, J and Fesharaki, M., 1995, "CO 2 -Retention Capability of CH 4 /O 2 -Fired Graz Cycle", Proceedings CIMAC 95, paper G07 Kwon, O. K., Pletcher R. H., Delaney R. A "Calculation of Unsteady Turbulent Boundary Layers" Journal of Turboachinery, April 1988, Vol. 110, pp Roe P. L Approximate Riemann Solvers, Parameter Vectors and Differencing scheme, Journal of Computational Physics 43, Sanz, W., Gehrer, A.,Paßrucker, H "An Implicit TVD Upwind Relaxation Scheme for the Unsteady 2D-Euler-Equations" ASME Paper 95 - CTP - 71 Woisetschläger, J., Jericha, H., Sanz, W. and Gollner F., 1995, "Optical investigation of transonic wall-jet film cooling", ASME Paper 95-CTP-26 Woisetschläger J., Jericha H., Sanz W., Pirker H. P., Seyr A., Ruckenbauer T Experimental Investigations of Transonic Wall-Jet Film Cooling in a Linear Cascade, European Conference - Antwerpen - March 97 REFERENCES Ameria, M. and Dybbs, A., 1993, "Coanda ejector - why it works", Proceedings of the SPIE, Vol 2052, pp Anderson, W. K., Thomas, J. L., Van Leer B "A Comparison of finite Volume Flux Vector Splittings for the Euler Equations" AIAA Paper No Arnone, A., Swanson, R.C "A Navier Stokes Solver for Turbomachinery Applications" Journal of Turbomachinery, April 1993, Vol. 115/ Arnone, A., Pacciani, R "IGV-Rotor Interaction Analysis in a Transonic Compressor Using the Navier Stokes Equations", ASME paper 96 - GT Fernholz, A. and Wille, R., 1971, " Grenzschichten und Wandstrahlen an stark gekrümmten Wänden (COANDA-Effekt)", DLR Forschungsbericht, FB71-46 Furukawa, M., Yamasaki, M, Inoue, M, 1992 "A Zonal Approach for Navier Stokes Computations of Compressible Cascade Flow Fields Using a TVD Finite Volume Method", Journal of Turbomachinery, Oct. 1992, Vol. 113/ Gehrer A., Paßrucker H., Jericha H., Lang J., 1996 Blade design and Grid generation for Computational Fluid Dynamics (CFD) with Bézier-curves and Bézier-surfaces, European Conference - Antwerpen - March 97, Paper No. 54 Gilchrist, A.R. and Gregory-Smith, D.G., 1988, "Compressible Coanda wall jet: predictions of jet structure and comparison with experiment", International journal of Heat and Fluid Flow, Vol.9, pp Gregory-Smith D.G. and A.R.Gilchrist, 1987, "The compressible Coanda wall jet - an experimental study of jet structure and breakaway", Heat and Fluid Flow, Vol. 8, pp Gregory-Smith D.G. and Hawkins,M., 1991, "The development of an axisymmetric curved turbulent wall jet", International Journal of Heat and Fluid Flow, Vol.12, pp

10 Zusammenfassung 181 Zusammenfassung In dieser Arbeit wurde ein Zeitschrittverfahren auf Finite-Volumen-Basis zur Berechnung der dreidimensionalen, kompressiblen, reibungsbehafteten und turbulenten Strömung in Schaufelgittern thermischer Turbomaschinen vorgestellt. Ausgehend von der integralen Form der Erhaltungsgleichungen wurde die Diskretisierung der konvektiven Terme entweder mit einem TVD-Upwindverfahren oder, alternativ dazu, mittels eines zentralen Verfahrens abgeleitet. Die Diffusionsflußbilanzierung erfolgte, -wie allgemein üblich- mit einem zentralen Verfahren. Die Modellierung der turbulenten Schwankungsbewegung erfolgte durch drei verschiedene Wirbelviskositätsmodelle (0-Gleichungs-, 1-Gleichungs-, 2-Gleichungsmodell). Zur zeitlichen Integration der Erhaltungsgleichungen wurde, unabhängig von der räumlichen Diskretisierung, ein explizites Vierschritt Runge Kutta Verfahren oder ein voll impliziter Zeitschrittalgorithmus verwendet. Für stationäre Berechnungen konnte die Konvergenz durch die Verwendung lokal veränderlicher Zeitschritte und durch die gleichzeitige Verwendung unterschiedlich feiner Rechengitter (MultiGrid) beschleunigt werden. Das Berechnungsgebiet wurde in mehrere strukturierte Rechennetze unterteilt (Multiblock), die wiederum einer vom Benutzer beliebig vorgegebenen Netzbewegung und Netzverformung unterliegen können. Die Generierung dieser Multiblocknetze erfolgte mit algebraischen Methoden, basierend auf Bézier-Kurven und Flächen, die zusätzlich mit einem differentiellen Netzgenerierungsverfahren zur Vergleichmäßigung von Unstetigkeiten kombiniert werden können. Die Validierung des entwickelten Rechenprogrammes erfolgte zunächst mit reibungsfreien oder laminaren Strömungsproblemen, deren analytische Lösung bekannt ist und die möglichst gezielt die einzelnen Programmodule abtesten sollten. Die Simulation der reibungsfreien Strömung in einer Lavaldüse und in einem Stoßwellenrohr zeigte, daß die bei transsonische Strömungen auftretenden Diskontinuitäten durch das Rechenverfahren sehr genau erfaßt werden. Weiters zeigte das Verfahren im Falle des ersten Impulsgitters von Hobson, bei einem schwingenden Plattengitter und bei der Berechnung eines rotierenden Gefäßes sowie im Falle der laminaren ebenen Plattengrenzschicht hervorragende Übereinstimmung mit der Analytik. Die Vor- und Nachteile der implementierten Zeitschrittverfahren zur Berechnung instationärer Vorgänge wurden anhand einer ebenen, laminaren Wirbelstraße mit Re 1 =100 diskutiert, wobei das explizite Verfahren dem impliziten an Genauigkeit deutlich überlegen ist, dessen Stärken allerdings wiederum in der Robustheit und in einer geringeren Gesamtrechenzeit lagen. Weiters wurden ebene turbulente Strömungen im Vergleich mit Meßdaten zur Verifizierung der Turbulenzmodellierung herangezogen, wobei im Falle der ebenen, turbulenten Plattengrenzschicht alle Turbulenzmodelle die Meßdaten gut widergaben. Im Falle von transsonischen Tragflügelströmungen traten allerdings deutliche Unterschiede auf, wobei das verwendete Eingleichungsmodell am genauesten war. Die Untersuchung des Nachlaufs hinter einem Turbinengitter zeigte, daß sowohl seitens der Numerik, als in Bezug auf die Meßtechnik weitere Studien zu diesem Thema notwendig sind. Die Berechnung des Wärmeübergangs an einer transsonischen Leitschaufelkaskade zeigte, daß die Problematik der Transitionsvorhersage von entscheidender Bedeutung ist. Die besten Ergebnisse konnten hier mit dem Zweigleichungsmodell unter zusätzlicher Verwendung einer modifizierten Formulierung des Produktionsterms für die turbulente kinetische Energie erzielt werden.

11 Zusammenfassung 182 Weitere 3D-Strömungssimulationen einer transsonischen Turbinenkaskadenströmung mit Reibung und Turbulenz zeigten im Vergleich mit Meßdaten, daß das verwendete Programmpaket in der Lage ist, die bei Turbomaschinenströmungen relevanten Sekundäreffekte zu simulieren. Im Anschluß wurden noch 3D-Stufenberechnungen zur aerodynamischen Auslegung der transsonischen Versuchsturbinenbeschaufelung des Institutes für thermische Turbomaschinen der TU-Graz durchgeführt. Dabei zeigten die Berechnungen gute Übereinstimmung mit den bereits von Paßrucker, 1997 durchgeführten Berechnungen. Weiters wurde die Durchströmung einer aus maschinendynamischen Gründen modifizierten Leitschaufel berechnet. Zum Abschluß ergaben die im Rahmen des Turbinenschaufelkühlungsprojektes des ITTM durchgeführten 2D-Strömungsberechnungen, daß Überschallstrahlen sich an gekrümmte Wände anlegen und damit Eigenschaften besitzen, die zur Turbinenschaufelkühlung im Bereich der Schaufelnase ausgenützt werden können.

12 Literaturangaben 183 Literaturangaben Arnone, A., Swanson, R.C., 1993, "A Navier-Stokes Solver for Turbomachinery Appliations", ASME Journal of Turbomachinery, Vol. 115, pp Arnone, A., Pacciani, R., 1996 "IGV-Rotor Interaction Analysis in a Transonic Compressor Using the Navier Stokes Equations", ASME paper 96 - GT Artner, W., 1997 "Implementierung eines Eingleichungsturbulenzmodells in einen 2D-Navier- Stokes Code", Diplomarbeit am Institut für Thermische Turbomaschinen und Maschinendynamik, TU-Graz Arts, T., Lambert de Rouvroit, M., Rutherford, A. W., 1990 Aero-thermal Investigation of a Highly Loaded Transonic Linear Turbine Guide Vane Cascade, Technical Note 174, von Karman Institute for Fluid Dynamics, Belgium Arts, T., 1994 Highly Loaded Transonic and Film Cooled Linear Turbine Guide Vane Cascade, in Numerical Methods for Flow Calculation in Turbomachines, VKI Lecture Series, Baehr, S., 1996, "Wärme- und Stoffübertragung", 2. Auflage, Springer-Verlag, ISBN Baldwin, B.S., Barth, T.J, 1990, "A One-Equation Turbulence Transport Model for High Reynolds Number Wall-Bounded Flows", NASA TM Baldwin, B.S., Lomax H., 1978, "Thin layer approximation and algebraic model for separated turbulent flows AIAA Paper Bassi, F., Osnagi, C., Perdichizzi, A., Savini, M., 1989, "Secondary Flows in a Transonic Cascade: Comparison Between Experimental and Numerical Results", Journal of Fluids Engineering, Dec. 1989, Vol. 111, p Bassi, F., Savini, M., 1992, "Secondary Flows in a Transonic Cascade: Validation of a 3-D Navier Stokes Code ", ASME-Paper 92-GT-62 Beam, R.M., Warming, R.F., 1978, "An Implicit Factored Scheme for the Compressible Navier-Stokes Equations", AIAA Journal, Vol. 16, No. 4 Benetschik, H., 1991, "Numerische Berechnung der Trans- und Überschall-Strömung in Turbomaschinen mit Hilfe eines impliziten Relaxationsverfahrens", Dissertation, RWTH Aachen Biswas, D., Fukuyama, Y., 1994, "Calculation of Transitional Boundary Layers With an Improved Low Reynolds Number Version of the k-ε Turbulence Model, Journal of Turbomachinery, Oct. 1994, Vol. 116, pp Van Den Braembussche, R. A., Leonard, O., Nekmouche, L., 1989, "Subsonic and Transonic Blade Design by Means of Analysis Codes", paper presented at the 64 th FDP 183

13 Literaturangaben 184 specialists' meeting on " Computational Methods for Aerodynamic Design (inverse) and Optimization", May 22-23, 1989, Loen, Norway Celigoj, C., 1990, "Festigkeitslehre", Vorlesungsskriptum, Technische Universität Graz Chakravarthy, S.R., 1988, "High resolution upwind formulations for the Navier - Stokes Equations", Lecture Series , Computational Fluid Dynamics Erhard, J., 1998, "Konstruktion, Aufbau und Kennfeldmessung der Versuchsturbine TTM", Dissertation, TU-Graz, 1998 Forstner, M., 1997, "Messung der Strömung durch ein transsonisches Turbinenschaufelgitter", Diplomarbeit an der Fakultät für Maschinenbau der TU Graz Furukawa, M., Yamasaki, M, Inoue, M, 1992 "A Zonal Approach for Navier Stokes Computations of Compressible Cascade Flow Fields Using a TVD Finite Volume Method", Journal of Turbomachinery, Oct. 1992, Vol. 113/ Furukawa, M., Nakano, T., Inoue, M., 1992 "Unsteady Navier Stokes simulation of transonic cascade flow using an unfactored implicit upwind relaxation scheme with inner iterations", Dept. of Mech. Eng. for Power, Kyushu Univ. Fukuoka, Journal of Turbomachinery Vol. 114 Gallus, H., E., Benetschik, H., Lohmann, A., Lücke, J., 1995, "Entwicklung eines Verfahrens zur Berechnung der reibungsbehafteten Strömung im Verdichtereinzelgitter", Abschlußbereicht zum Forschungsvorhaben der Arbeitsgemeinschaft Hochtemperatur Gasturbine, Vorhabengruppe 1.1.2: "Berechnung der mehrdimensionalen reibungsbehafteten Strömung in Schaufelgittern und Turbomaschinen", Institut für Strahlantriebe und Turboarbeitsmaschinen der RWTH Aachen Gehrer, A., 1994, "Numerische Berechnung der Turbomaschinenströmung mit Hilfe eines TVD-Upwind Verfahrens", Diplomarbeit an der Fakultät für Maschinenbau der TU Graz Giles, M.B., 1990, Non-reflecting Boundary Conditions for Euler Equation Calculations, AIAA Journal, Vol.28, No.12, 1990 Godunov, S.K, 1959, "A Finite-Difference Method for the Numerical Computation of Discontinuous Solutions of the Equations of Fluid Dynamics", Matematicheskii Sbornik, Vol. 47, S , 1959 Gretler, W. 1987, "Strömungslehre", Vorlesungsskriptum, Technische Universität Graz Gretler, W. 1990, "Wärmeübertragung", Vorlesungsskriptum, Technische Universität Graz Hellsten, A., 1996, Implementation of a One Equation Turbulence Model into the FINFLO Flow Solver, Helsinki University of Technology, Laboratory of Aerodynamics, Report N. B- 49, Series B, 1996 ISBN

14 Literaturangaben 185 Hirsch, C., 1989, "Numerical Computation of Internal and External Flows", Vol.1+2, Wiley & Sons, 1989 Hopkins, E., J., Inouye, M., 1971, "An Evaluation of Theories for Predicting Turbulent Skin Friction and Heat Transfer on Flat Plates at Supersonic and hypersonic Mach Numbers", AIAA Journal, June 1971 Hobson, D.E., "Shock-free transonic flow in turbomachinery cascades", University of Cambridge, Dept. of Eng. Report CUED/A Turbo/TR 65 Hoschek, J., Lasser, D., 1989, "Grundlagen der geometrischen Datenverarbeitung", B. G. Teubner Stuttgart, 1989 Jameson, A., Schmidt, W., Turkel, 1981, "Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge - Kutta Time - Stepping Schemes", AIAA 14th Fluid and Plasma Conference Palo Alto Jameson, A., 1983, "The Evolution of Computational Methods in Aerodynamics", ASME Journal of Applied Mechanics, Vol. 50, pp Jameson, A., Yoon, S., 1986, "Multigrid Solution of the Euler Equations Using Implicit Schemes", AIAA Journal, Vol. 24, No. 11, Nov. 1986, p Kato, M., Launder, B.E., 1993, "The modelling of turbulent flow around stationary and vibrating square cylinders. In Proc. 9th Symposium on Turbulent Shear Flows, Kyoto, pages , August Kindlhofer, H., 1990, "Numerische Erstellung gekrümmter Rechennetze", Diplomarbeit, Technische Universität Graz, 1990 Kloker & Borsboom, 1988, "A Fully Implicit Scheme For Unsteady Flow Calculations Solved by the Approximate Factorization Technique", Lecture Series , Computational Fluid Dynamics Koiro, M., Lakshminarayana, B., 1998, 'Simulation and Validation of Mach Number Effects on Secondary Flow in a Transonic Turbine Cascade Using a Multi-Grid, k-e Solver' Journal of Turbomachinery, 1998, Vol. 120, p Larsson, J., 1996, "NUMERICAL SIMULATION OF TURBINE BLADE HEAT TRANSFER, Thesis for the degree of Licentiate of Engineering, Department of Thermo and Fluid Dynamics Chalmers, University of Technology Lefebvre, M., Arts, T., 1997, "Numerical Aero-Thermal Prediction of Laminar/Turbulent Flows in a Two-Dimensional High Pressure Turbine Linear Cascade, European Conference - Antwerpen - March 97 Lücke, J.R., 1997, "Turbulenzmodellierung zur Berechnung abgelöster Strömungen in Turbomaschinen", Dissertation an der RWTH Aachen, D82, Shaker Verlag, Aachen 185

15 Literaturangaben 186 Lukasser, R.F., 1996, "Festigkeits- und Schwingungsrechnung von Laufschaufel und Laufradscheibe der Versuchsturbine des Institutes für Thermische Turbomaschinen und Maschinendynamik, TU-Graz", Diplomarbeit am Institut für Thermische Turbomaschinen und Maschinendynamik, TU-Graz Mayerhofer, J., 1998, "Berechnung instationärer Strömungen durch schwingende Schaufelgitter", Diplomarbeit am Institut für Thermische Turbomaschinen und Maschinendynamik, TU-Graz Migliorini, F., Michelassi, V., 1997, "Transition and Heat Transfer Modelling in Transonic Linear Cascade, European Conference - Antwerpen - March 97 Engquist, B., Osher, S., 1980, "Stable and entropy satisfying approximation for transonic flow calculations", Mathematics of Computation, Vol. 34, pp Paßrucker, H., 1997, "Numerische Berechnung der 3D-Eulerströmung durch transsonische Turbomaschinenstufen", Dissertation an der Fakultät für Maschinenbau der TU Graz Perdichizzi, A., Uribaldi, M., Zunino, P. 1989, "Secondary flows and Reynolds stress distributions downstream of a turbine cascade at different expansion ratios", AGARD-CP 469, Ref. 6 Perdichizzi, A., 1990, "Mach Number Effects on Secondary Flow Development Downstream of a Turbine Cascade", ASME Journal of Turbomachinery, Vol. 112, October 1990 Prandtl, L., 1925, "Bericht über Untersuchungen zur ausgebildeten Turbulenz", Zeitschrift für angewandte Mathematik und Mechanik, Vol. 5 Press, H., Teukolsky, S. A., Vetternig, W. T., Flannery B. P., 1989 "Numerical Recipes", Cambridge Press Pulliam, T.H., 1986, "Artificial Dissipation Models for the Euler Equations", AIAA Journal, Vol. 24, No. 12 Roe, P. L. 1981, "Approximate Riemann Solvers, Parameter Vectors, and Differencing schemes", Journ. of Computational Physics Rumsey, L., Vatsa, V.N. 1993, "A Comparison of the Predictive Capabilities of Several Turbulence Models using Upwind and Central-Difference Computer Codes, AIAA Sanz, W, 1993, "Numerische Berechnung der Transschall- und Überschallströmung in Thermischen Turbomaschinen", Dissertation, Technische Universität Graz Schröcker, H.P., 1998, "Die Konstruktion orthogonaler Fortsetzungen von Bezier-Kurven und -Flächen", Diplomarbeit am Institut für Geometrie, TU-Graz Schuemie, H.A., 1998, "Dreidimensionale Berechnung der Turbomaschinenströmung mit Hilfe eines Finite-Volumen Verfahrens", Diplomarbeit am Institut für Thermische Turbomaschinen und Maschinendynamik, TU-Graz 186

16 Literaturangaben 187 Sieverding, C., 1976, "Transonic Flows in Axial Turbomachinery", VKI Lecture Series 84 Siikonen, T., 1991, "A Three-Dimensional Multigrid Algorithm for the Euler and the Thin- Layer Navier-Stokes Equations", Helsinki University of Technology, Faculty of Mechanical Engineering, Report No. A-12, Series A Siikonen, T., 1994, "An Application of Roe's Flux-Difference Splitting for k- ε Turbulence Model", Helsinki University of Technology, Faculty of Mechanical Engineering, Report No. A- 15, Series A Spalart, P. R., Allmaras, S. R., 1994, "A One Equation Turbulence Model for Aerodynamic Flows, La Recherche Aerospatiale, No.1, pp 5-21 Vandromme, D., 1991, "Turbulence Modelling for Compressible Flows and Implementation in Navier-Stokes Solvers", VKI Lecture Series Whitehead, D.S., 1987, "Classical Two-Dimensional Methods", AGARD Manual on Aeroelasticity in Axial-Flow Turbomachines, Unsteady Turbomachienery Aerodynamicsm Vol 1., Chap. 2, M.F. Platzer and F.O. Carta, eds., AGARD-AG-298 Wohlhart, K., 1988, "Hydromechanik", Vorlesungsskriptum, Technische Universität Graz 187

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