Mathematical Modelling of Transonic Flows in 2D Cascade

Mathematical Modelling of Transonic Flows in2D CascadeJ. Fořt J. Fürst K. Kozel P. PunčochářováAbstractThe work deals with mathematical modelling and numerical solutionof transonic flows in a 2D cascade near the so-called choked regime using3 basic models: potential, inviscid, and viscous flows. We consideralso several possible formulations of upstream boundary conditions forthree types of 2D cascades: a turbine cascade, a compressor cascade andDCA 8% cascade. The numerical results for transonic flows in DCA 8%cascade with M ∞ > M ch (Mach number of choked flow) are compared toexperimental results of IT AS CR.1 Mathematical modelsThree mathematical models are used to describe transonic flows: potentialmodel, inviscid model described by the system of Euler equations and viscousmodel described by the system of Navier–Stokes equations either for laminarflow or in RANS formulation for turbulent flows. The potential model was consideredboth in full form and in simplified version known as a small disturbancepotential model.Full potential equation is steady quasilinear partial differential equation ofsecond order and changes its type (elliptic-parabolic-hyperbolic type) accordingto local Mach number. The unsteady system of Euler equations is nonlinearsystem of first order partial differential equations of hyperbolic type. The systemof Navier–Stokes equations is nonlinear parabolic system of second order. Asobvious, the boundary conditions have to be chosen according to the type ofequations.2 Choked flowsThe problem of choked flows for compressible flows is well known in 2D. Fromquasi-1D theory follows, that the mass flow ṁ is a function of inlet Mach numberM ∞ and that there exists maximal mass flow given by the geometry of thechannel [2]. Therefore ṁ(M ∞ ) ≤ ṁ max . Denote by M ch1 and M ch2 two rootsof ṁ(M ∞ ) = ṁ max . Then from quasi-1D analysis follows that there are no 2Dflows with upstream Mach number M ∞ ∈ (M ch1 , M ch2 ).It is possible to realize flows for upstream Mach number M ∞ < M ch1 < 1(Mach number of choked flow) on the other hand the solution for M ∞ > M ch1is not obvious. The situation is following:

• for turbine cascades one can realize flows with M ∞ < M ch1 with closedsonic line (i.e. going from one profile to another one, see fig. 1) and it isnot possible to increase M ∞ over M ch1 . Although it is possible to changethe flowfield in the outflow part by changing the outlet pressure p 2 (see[8], [10], [1], [7]).• for transonic compressor cascades there are flows with supersonic inletM ∞ > M ch2 > 1 (see fig. 2, [4]).• Mr. Dvořák (IT AS CR) proved experimentally (see [2]) the existence oftransonic flow in impulse turbine DCA 8% cascade with M ∞ ∈ (M ch1 ,M ch2 ) see fig. 3 and 4 showing flow field for M ∞ > M ch1 . Numericallywas this effect confirmed in [9] by numerical solution of small disturbancepotential equation and in [5], [6] using numerical solution of Euler equations.3 DCA 8% cascadeIn this chapter authors show numerical results of transonic flows through 2DDCA 8% 1 cascade compared to experimental results of IT AS CR for differentupstream Mach numbers. Authors confirmed the results of IT AS CR in thesense, that the flow exists also after first choked regime. Numerical results areobtained using potential model and the system of Euler equations for inviscidcompressible flows.3.1 Upstream boundary conditionsThe choice of proper upstream boundary conditions depends on the mathematicalmodel.For the case of full potential equation as well as for the small disturbance potentialequation one considers usually Dirichlet or Neumann boundary conditionbecause the equation is for M ∞ < 1 elliptic.Similar situation is for the set of Navier–Stokes equations which are parabolicand therefore it is again possible to prescribe Dirichlet or Neumann conditions.On the other hand, the set of Euler equations is hyperbolic and the boundaryconditions should be chosen according to the sign of eigenvalues of Jacobiansof fluxes. In upstream part with M ∞,n = M ∞ cos β < 1 (M ∞,n is the normalMach number computed as a ratio of the magnitude of velocity componentnormal to inlet boundary and local speed of sound; β is the angle betweeninward normal and inlet velocity vector) one eigenvalue is negative and theothers are positive. It means that for correct formulation in 2D three quantitiesshould be prescribed and one should be extrapolated. However, the choice of thequantities is somewhat problematic. Consider transonic flows through DCA 8%cascade. In order to compare our numerical results with the experimental dataobtained at IT AS CR, we have to do the calculation with given inlet Machnumber. Therefore we consider M ∞ as a given quantity and we extrapolate otherquantity. Another possibility is to keep e.g. inlet angle and stagnation pressureand stagnation speed of sound and to extrapolate M ∞ . It is even possible to use1 DCA 8% profile is a blade composed of two circular arcs with relative thickness 8%.

Dirichlet conditions for all unknowns due to appearance of numerical viscosityterm in numerical scheme.Question is what kind of boundary condition is better for the case of comparisonwith experimental data. When we keep given M ∞ , then the solution isclose to the experimental data. More correct approach with the extrapolationof M ∞ does not allow to do the computation with given M. However, it is stillpossible to change the value of M ∞ by changing e.g. inlet angle.3.2 Numerical simulation of flows through DCA 8% cascadeWe considered here flows through DCA 8% cascade with upstream Mach numberclose to M ch and inlet angle α 1 ≈ 0. Figures 5, 6 show the isolines of Machnumber for different upstream Mach number and different models and boundaryconditions. The interesting thing is the change of upstream inlet angle forM ∞ > M ch in the case of upstream Dirichlet conditions for all variables. Theangle changes to the value α ≈ 5.125 o . When we consider given p 0 , ρ 0 and α 1and we extrapolate M ∞ , we have to iterate the inlet angle α 1 in order to obtainflows with M ∞ equal to a desired value. In fact, the inlet Mach number M ∞is for the case of choked flow a function of inlet angle α. After few iterationwe obtain corresponding inlet angle α 1 ≈ 5.125 o (practically the same value asabove).This change of upstream angle of attack explains the existence of flows afterfirst “choking” conditions in the DCA 8% cascade (see fig. 7).4 Concluding remarksThis article shows several numerical results of choked flows through DCA 8%cascade obtained by several finite volume schemes and using several formulationsof upstream boundary conditions. The numerical simulation explains thepossiblity of flows with M ∞ > M ch . Those flows were observed also experimentallyat the IT AS CR. The second problem mentioned in this article is theproper formulation of upstream boundary condition for flows through DCA 8%cascade described by the system of Euler equations.Acknowledgement: This work was partly sponsored by the grant GA ČRNo. 201/05/0005 (Kozel, Fořt, Punčochářová) and research plan of the Ministryof Education of the Czech Republic No. 6840770010 (Fürst).References[1] J. Dobeš, J. Fürst, J. Fořt, J. Halama, and K. Kozel. Numerical solutionof transonic flows in 2D and 3D axial and radial turbine cascades. InProceedings of 5th European conference on Turbomachinery, pages 1105–1114, Prague, 2003.[2] Rudolf Dvořák. On the development and structure of transonic flow incascades. In Proceedings of ”Symposium Transsonicum II”, pages 297–305.Springer-Verlag Berlin, 1976.

(a) Experimental result of IT AS CR (b) Numerical solution, Eulerequations, unstructured gridFigure 1: Choked flow in a turbine cascadeFigure 2: Choked flow in a compressor cascade [9]

Figure 3: Sonic line development [2][3] J. Fořt and K. Kozel. Numerické řešení transonickckého obtékání rovinnélopatkové mříže. Strojnický časopis, 35(3), 1984.[4] J. Fořt and K. Kozel. Numerical solution of inviscid two-dimensional transonicflow through a cascade. In ASME Paper, number 86-GT-19, 1986.[5] J. Fürst, K. Kozel, P. Punčochářová, and P. Šafařík. Transonic flowsthrough DCA 8% cascade. In K. Kozel J. Příhoda, editor, Proceedings of”Topical Problems 2005”, pages 61–64. IT CAS CZ, February 2005. ISBN80-85918-92-7.[6] Jiří Fürst. Numerical modeling of the transonic flows using TVD and ENOschemes. PhD thesis, ČVUT v Praze and l’Université de la Méditerranée,Marseille, February 2001.[7] Jan Halama, Tony Arts, and Jaroslav Fořt. Numerical solution of steadyand unsteady transonic flow in turbine cascades and stages. Computersand Fluids, 33:729–740, 2004.[8] Kozel K., Louda P., and Příhoda J. Numerical solution of transonic turbulentflow through a turbine cascade. In J. Příhoda K. Kozel, editor,Proceedings of Conference “Topical Problems of Fluid Mechanics”, pages73–76, Praha, 2004. IT CAS CZ. ISBN 80-85918-86-2.[9] K. Kozel, J. Polášek, and M. Vavřincová. Numerical solution of transonicflow through a cascade with slender profiles. In Lecture Notes in Physics.Springer Verlag, 1978.[10] P. Šafařík, M. Štastný, and M. Babák. Numerical and experimental testingof transonic flow in the etalon turbine cascade SE 1050. In G. BoisM. Štastný, C. H. Sieverding, editor, 5th European Conference on Turbomachinery.Czech Technical University in Prague, 2004.

(a) M ∞ = 0.813, α ∞ = 0 ◦ , p 2 /p ∞ = 1 (b) M ∞ = 0.832, α ∞ = 0 ◦ , p 2 /p ∞ = 1(c) M ∞ = 0.849, α ∞ = 0 ◦ , p 2 /p ∞ = 1 (d) M ∞ = 0.863, α ∞ = 0 ◦ , p 2 /p ∞ = 1(e) M ∞ = 0.946, α ∞ = 0 ◦ , p 2 /p ∞ = 1 (f) M ∞ = 0.982, α ∞ = 0 ◦ , p 2 /p ∞ = 1(g) M ∞ = 1.013, α ∞ = 0 ◦ , p 2 /p ∞ = 1 (h) M ∞ = 1.073, α ∞ = 0 ◦ , p 2 /p ∞ = 1Figure 4: Transonic flow development in DCA 8% cascade, experimental resultsof IT AS CR [2]

(a) M ∞ = 0.845, α ∞ = 0 ◦ , p 2 /p ∞ = 1 (b) M ∞ = 0.850, α ∞ = 0 ◦ , p 2 /p ∞ = 1(c) M ∞ = 0.930, α ∞ = 0 ◦ , p 2 /p ∞ = 1 (d) M ∞ = 0.950, α ∞ = 0 ◦ , p 2 /p ∞ = 1(e) M ∞ = 1.05, α ∞ = 0 ◦ , p 2 /p ∞ = 1 (f) M ∞ = 1.08, α ∞ = 0 ◦ , p 2 /p ∞ = 1(g) M ∞ = 1.11, α ∞ = 0 ◦ , p 2 /p ∞ = 1 (h) M ∞ = 1.15, α ∞ = 0 ◦ , p 2 /p ∞ = 1Figure 5: Transonic flow development in DCA 8% cascade, calculation usingcomposite scheme with 40 LW steps followed by one LF step and a mesh with150 × 30 cells [5]

(a) α 1 = 2.5 ◦ , p 2 = 0.48p 0 , (M 1 = 0.863) (b) α 1 = 4.0 ◦ , p 2 = 0.48p 0 , (M 1 = 0.946)(c) α 1 = 4.6 ◦ , p 2 = 0.48p 0 , (M 1 = 0.982) (d) α 1 = 5.2 ◦ , p 2 = 0.48p 0 , (M 1 = 1.013)Figure 6: Transonic flow development in DCA 8% cascade, calculation usingmodified Causon’s scheme and a mesh with 105 × 30 cells [6](a) Dirichlet conditions for all variables, (b) Extrapolation of inlet pressure,α 1 = 1 o α 1 = 5.17 oFigure 7: Two variants of inlet conditions for DCA 8% cascade with inlet Machnumber M 1 = 1.08 and p 2 = 0.48p 0