Application of Analytical and Numerical Methods for the Design of Transonic Axial Compressor tages rof. Dr.-Ing. Martin Böhle Mechanical Engineering University of Wuppertal Gaußstraße 0, 4097 Wuppertal Germany Abstract: Usually, the first stage of a compressor of a gas turbine works in the region of transonic flow, so that the designer has to take into account a lot of flow details like s, boundary layer separation and their interaction, when a new machine is defined. Beside other problems the choice of the blade geometry and profile family is one important consideration. For the design of the blades analytical and numerical methods are applied, but there is always a difference in their results, because the analytical methods are based on simplified assumptions and the numerical techniques make use of turbulence models. o the question arises: How large is the difference between the analytical and numerical results? This is theme of the following contribution. Key-Words: Transonic Flow, Analytical and Numerical Calculation Techniques, hocks, Unique-Incidence-Condition Introduction In order to achieve large pressure ratios with not many stages the first or the first and the second stage of an axial compressor work in the transonic flow region. Near the hub the relative flow can be subsonic, but at the tip of the rotor blades the relative flow is supersonic, so that the character of the flow changes from high subsonic to small supersonic relative Mach number flow over the radius. Although occurring flow details like s, boundary layer flows and interactions of s and boundary layers are understood, it is still a problem to predict the performance of a rotor blade, because analytical calculation techniques are based on simplified assumptions. On the other hand, numerical tools are limited to resolution, turbulence modelling problems etc., so that the results of a three dimensional transonic flow simulation must not agree with the real flow. Especially, the design process is difficult. There exist no common design procedure for rotors working in the transonic flow region []. Nevertheless, analytical and numerical tools find their applications for the design process and the questions arises how large the differences of the analytical and numerical results are. At present special attention is given to numerical procedures, which are available as commercial versions on the market. They find increasing attraction for applications for all kind of flows of industrial interest. The transonic flow inside a compressor is also modelled by such numerical techniques more and more during the design process. Although commercial, numerical procedures are easy to handle and to integrate into a design process, they do not belong to the classical group of design tools, because the corresponding CU-time is too large. On the other hand, analytical tools work more quickly, but -as aforementioned- are based on simplified assumptions and as a rule do not make it possible to model the whole flow field. When analytical and numerical tools are applied for a design of a transonic rotor, then it is of interest to know how large the differences of analytical and numerical results are. This is the major question of the present contribution. First, a short description is given about the flow inside a transonic compressor and the design process (for details s. [], []). This is followed by an explanation of the flow details occurring in rotor blades. Then two analytical prediction methods are introduced. The first is applied for the calculation of the unique-incidence and the second one is used for performance analysis. The results of the analytical tools are compared with numerical results, which are produced by the application of FLUENT. Flow through transonic compressor Fig. shows the merdional cross section of a compressor. The flow relative to the casing is subsonic, but due to the rotation of the rotor the flow velocity
relative to the rotor blades is larger than the speed of sound, i.e. the flow is supersonic. The rotor blades turn the flow and as a rule the relative flow behind the rotor is subsonic and the pressure of the flow increases from the inlet to the outlet region of the blades. o there is a change from supersonic to subsonic flow. In Fig. a sketch of the transonic flow is depicted. In a very large distance upstream of the cascade, the Mach number is larger than one and the flow angle is β. Further downstream s running through the flow field, which are generated by the nearly sharp leading edge of the profiles. The extent of these s acts along a very large distance upstream, until they degenerate into a Mach wave. A 3 A stages A second the so named passage - occurs between two blades. This is also generated at the leading edge and the Mach number of the flow changes from supersonic to subsonic over the passage. Rotor tator M > w w flow relative to rotor cross section A-A u W λ β axial direction l passage M> M< t c flow relative to casing boundary layer Figure : Flow Conditions for a tage of a Transonic Compressor. In Fig. a cross section of the rotor blades is depicted. When a rotor of a compressor is designed, many cross sections from the hub to the tip are considered (for example 0). o the special emphasis is given to the flow through cascades consisting of supersonic profiles as shown in Figure. This is the subject of the following contribution, i.e. to calculate the subsonic/supersonic flow through such cascades by analytical and numerical methods and to compare their results. If the flow is supersonic and this is the case for the first stage of a transonic compressor- the flow is difficult to calculate. The reason for these difficulties are: upersonic flows contain and they cause additional flow losses. The flow is choked, i.e. the axial component of velocity W is not influenced by the downstream pressure (if the pressure is not too large). Figure : ketch of the flow through a supersonic cascade. As aforementioned the flow is choked i.e. the axial component of W is constant -, furthermore the angle β is determined by the axial component and the circumferencial velocity U. For design purposes it is of interest to calculate this angle β, which is dependent of the inlet Mach number M. The dependence of the flow angle β is often called as unique-incidencecondition. In the next chapter an analytical calculation method for β (M ) is explained, which is described in detail in [3]. Furthermore an analytical method for the prediction of the performance curve of a cascade is introduced (for details s. [8]) and the results of both analytical methods are compared with numerical results, which are produced by the FLUENT-Code.
3 Analytical methods 3. Calculation of the flow angle β There are several calculation methods for the flow angle for β (M ), s. for comparison [4], [5], [6]. The following methods stems from [3]. The geometry of the cascade is given (s. Fig. 3), i.e. the stagger angle λ, the solidity l/t and the shape of the profiles as y=f(x). M > w For the calculation it is assumed that the flow is isentropic (only weak s, no friction) the is attached to the profile. the leading edge is sharp. the flow relative to the casing is subsonic. In Fig. 3 points and are marked. ont is located just in front of the. From point a left running characteristic runs to point, so that in point and the flow conditions are equal, i.e. it can be stated M = M () β = () β It makes sense to assume that the Mach number M is given in point and to calculate the coordinates of point. In a further step the inlet Mach number M is determined. For the calculation of point the following equations are valid: β = λ + ν tan( λ β y t cosλ y (3) α + ν ) = (4) t sin λ x dy ν = (5) dx Figure 3: Geometry of the cascade. l β s x t sin α = = (6) M M Equation (3) and (4) are based on simple geometrical considerations. α stands for the Mach angle. Now the procedure is as follows. It is assumed that the Mach number M is known. Then it is possible to use equations (4), (5) and (6) for the determination of the coordinates (x, y ) of point. Along the characteristic the flow condition is constant and for the mass flow through this characteristic can be stated m& = ρ w sin α (7) The mass flow of the inlet flow can be expressed as w t cos m& = ρ β (8) and by equating equation (7) with equation (8) and using some simple basic gasdynamical relations yields equation (9): F( M ) t cos β F( M ) sinα = (9) c T c T F(M) p 0 p 0 A κ+ κ κ ( κ ) = M + M (0) κ c p and κ stand for the specific heat and ratio of specific heats, respectively. F(M) is the compressible mass flow function. In equation (9) the unknowns are M and β. A further equation for the calculation of the unknowns is given by the well-known randtl-meyer relation: β + θ M ) = β + (M ) () ( θ θ(m) stands for the randtl-meyer function, which is described in nearly all text books about gasdynamics. 3. Calculation of the performance curve Freemann and Cumpsty describe in [8] an analytical method for the prediction of supersonic compressor blades. The methods is based on the conversation of mass, momentum and energy. The conversation laws are applied for a control volume shown in Fig. 4. It is assumed that the pressure and loss increase mainly takes place in the region of the leading edge. Furthermore the calculation is based on the assumptions that
the blades are lightly cambered in the leading edge region the blades are thin there exist a blade direction, which is characterised by the stagger angle λ The three conversation equations are ρ W g cos β = ρ W ( g t) cosλ () p M > = p g cosλ + p g cosλ + p κ M g cosβ cos( λ β) (3) M (g t) cosλ κ κ κ T + M = T + M (4) With some algebraic manipulation of equations (), (3) and (4) the following equation can be derived. t + κ M g κ t + M M g cosλ + κ M cos( β λ) cosβ = κ + M M g M < Figure 4: Conversation laws are applied for a control volume. (5) The left hand side is a function of M, t/g and the right hand side is a function of M, β. If the geometry of the cascade, the inlet Mach number M and flow angle β are given, equation (5) can be used for the calculation of M. Then it is easy to determine the pressure ratio p /p and all values of interest by equations () to (3). t 4 Numerical method and results 4. Description of numerical method Numerical simulations are performed by the softwarepackage from FLUENT [9]. FLUENT stands for a numerical technique to solve the Reynolds-averaged equations for technical applications. The numerical method belongs to the group of the Finite Volume-Method and can be applied on structured and unstructured grids. A lot of turbulence models are implemented into the FLUENT-code. The results of the simulations shown in the present contribution are performed on an D-unstructured triangular grid. It was assumed that the flow is turbulent, and for modelling of the turbulence effects the k-εmodel was chosen. In Fig. 5 the grid is shown for a cascade consisting of DCA-profiles (Double-Circular-Arc), which are applied for transonic compressor flows for Mach numbers reaching from M =.0...3. The stagger angle of the cascade is λ=60, the solidity l/t =.5, the thickness ratio d/l =0.0 and the camber angle of the profiles is 0. In the numerical flow field seven profiles are integrated and periodic boundary conditions are set on the top and bottom side of the grid (not shown in Fig. 5). Figure 5: The grid shown for a cascade consisting of DCA-profiles. The flow moves from left to right. On the left hand side an pressure inlet boundary condition is set (total pressure, total temperature, flow direction and turbulence quantities), on the right hand side a pressure outflow boundary condition is imposed. A small static pressure is chosen in such a way that the upstream flow is not influenced by the outflow boundary condition.
The flow direction is varied and the inlet Mach number on the inlet boundary is calculated by the FLU- ENT-Code. The results are described below. β = 6,8 65,8 M =,,8 4. Numerical Results A result of a flow simulation is shown in Fig. 6. The inlet flow angle is β =64 and the Mach number M calculated by FLUENT is M =.8. β s M =, 9 s M =, 3 &KDUDFWHULVWLFV The first reaching from the leading edge upstream is not resolved by FLUENT (may be too weak). On the suction side of the profiles the flow is accelerated and at the trailing edge the passage occurs. The passage is oblique, so that there is no change from supersonic to subsonic flow. Figure 7: Variation of the inlet flow angle β in dependence of the Mach number M for a DCA-cascade (t/l=., λ=60 ). In Fig. 8 the variation of the aforementioned flow angle in dependence of the Mach number M for a range of solidity from l/t =0.833....5 is shown. Additionally the results of the flow simulations are marked. For small Mach numbers there is a good agreement between the analytical and numerical results. But if the Mach number gets larger the difference between both results increases, because the analytical results are based on the assumption that the flow is isentropic. M=. M=.3 M=.4 M=.5 M=.6 passage It is worth to mention that both the Mach number and the variation of the Mach number are small. FLU- ENT s code is sensitive to small variations of flow direction at the inlet boundary, although FLUENT is applicable for all kind of flows with small and large Mach numbers (for example M >0). But FLUENT s code need along time until the solution converges. Multigrid techniques must be switched off and the equations of fluid motion must be solved coupled with a small Courant number. Figure 6: Numerical results for cascade flow (λ=60, t/l=0.8, M =.8. The passage is not generated by the leading edge, because a randtl-meyer-expansion occur around the leading edge. In the following chapter the corresponding analytical results are described and compared with this numerical solution. 5 Analytical results In Fig. 7 the variation of the inlet flow angle β in dependence of the Mach number M for a DCA-cascade (t/l =., λ=60 ) is shown. In addition to that the variation of the location and extent of the characteristic are depicted. The difference between the smallest and largest flow angle is very small, so one can recognise that transonic flow is very sensitive to flow angle variations as it has always been understood. In Fig. 9 the result of equation (5) corresponding to the analysis described in [8] are shown. For all M > 66.5 66 65.5 65 β 64.5 64 63.5 63 Unique Incidence Condition d/l = 0.0 φ = 0 Numerical Results t/l=. t/l=0.8 t/l=.0 6.5...3.4.5.6.7.8.9 Mw Figure 8: Variation of the aforementioned flow angle in dependence of the Mach number M.
of the RH-value can be found a LH-value. o the result of this analysis does not agree with the numerical result and the unique-incidence-calculation. Figure 9: Analytical result corresponding to [8], DCA- rofile d/l=0.0, β =64, λ=60, t/l=0.8. The analysis described in [8] delivers acceptable results, but due to the simplifications used in [8] such a sensitive unique incidence relation β (M ) departs significantly from aforementioned results. By the application of the analysis described in [8] for M =.8 an angle β = 6.5 is calculated. References: [] N. A. Cumpsty, Compressor Aerodynamics, Addison Wesley Longman Limited, 989 [] R. I. Lewis, Turbomachinery erformance Analysis, John Wiley & ons Inc., 996 [3] A. Bölcs,. uter, Transsonische Turbomaschinen, G. Braun-Verlag, 986 [4]. Levine, The Two Dimensional Inflow Conditions for a upersonic Compressor with Curved Blades, Wright Air Development Center, 956 [5] H. tarken, Untersuchung der trömung in ebenen Überschallverzögerungsgittern, DFVLR FB 7-99, 97 [6] H.-J. Lichtfuss, H. tarken, upersonic Cascade Flow, DFVLR onderdruck Nr. 376 ergamon ress Ltd. Vol. 5,974 [8] C. Freemann, N.A. Cumpsty: A Method for the rediction of upersonic Compressor Blade erformance, AIAA Journal of ropulsion and ower, Vol. 8, No., pp. 99-08, 99 [9] FLUENT-Handbooks, Fluent Incorporated Lebanon, NH 03766, 998 6 Conclusions The results described above allow the following conclusions: For a design process of a transonic compressor both the analytical and numerical tools are indispensable. The performance of a supersonic cascade is very sensitive to flow angle and Mach number variations and neither the numerical nor the analytical methods can predict the flow exactly. The variation of the Mach number and flow angle is small. And there is only a small range for design. Analytical tools make it possible to get immediate results, but especially one must keep in mind the simplifications, on which the results are based. Numerical methods give information on the flow structure. But for the simulation of transonic flows weak s may be resolved insufficiently. The experience of engineers and the application of measurements are indispensable for the design process.