“Centre of research and experimental development of reliable energy production” TE01020068 of the Technology Agency of the Czech Republic

Lukeš, V.

Moravcová, F.

Rohan, E.

University of West Bohemia Digital Library

Sensitivity analysis for an almost periodic unsteady flow problem; application to turbo-machinery modelling

DSpace at University of West Bohemia

COMPUTATIONALCHANICSME November 4 - 6, 2019Srníconference with international participation35th2019Sensitivity analysis for an almost periodic unsteady flowproblem; application to turbo-machinery modellingF. Moravcováa, V. Lukeša, E. RohanaaNTIS – New Technologies for Information Society, Faculty of Applied Sciences, University of West Bohemia,Univerzitní 8, 301 00 Plzenˇ Czech Republic1. IntroductionThe shape optimization in both the compressible and incompressible flow problems still belongsto areas of intensive research due to its high applicability on one hand and the complicated math-ematical structure of the problem on the other hand. The present contribution is a continuationof our previous work [2], where the shape optimization of a turbine blade was reported for theincompressible 2D flows with the aim to minimize the dissipation power. There the steady in-compressible Navier-Stokes (N-S) system of equations was discretized using the finite volumemethod and the continuous version of the adjoint system method was employed to computetotal shape gradients of the objective function.In the present work, we are concerned with compressible 2D flows in channels whose the ge-ometries vary periodically with time. Such a situation arises in the stator-rotor configuration ofturbines where the mutual position of the blades commutes periodically. Obviously, the steadymotion the rotor blades which are arranged periodically induces the unsteady flows which canbe considered as quasi-periodic. Our study relies on this hypothesis, we assume the existenceof T -periodic solutions. The time period T = H/U is given by the circumferential geometricperiod H and the circumferential speed U associated with a given radius of the rectificationplane where the 2D flow problem is defined.As the new contribution, for a general shape optimization problem we introduce the stateproblem with an implicitly stated T -periodic condition and propose an iterative algorithm ofthe sensitivity analysis. In contrast with the standard evolutionary problems, the sensitivityanalysis (SA) of a time periodic problem leads to a more complicated computational scheme.For efficiency of the SA computations, we propose an iterative algorithm which is constructedfor the discretized state problem. We developed two versions of the SA algorithm according tothe two commonly used approaches, the Direct Differentiation and the Adjoint System methods[3, 4]. Numerical solutions of the state problem are obtained using the in-house developed CFDsoftware FlowPro [1] based on the Discontinuous Galerkin Method whereby the first-orderapproximation is used, thus, actually yielding the discretization scheme of the Finite VolumeMethod.2. The flow problemThe flow in the stator-rotor channels is represented by a reduced two-dimensional model. Itsgeometrical configuration reflects the periodic arrangement of the blades, see Fig. 1. The prob-lem is imposed in a domain constituted by two subdomains associated with the stator and rotor123Fig. 1. Mach distribution in the flow domain at various time steps of the simulation with α = 0 (attackangle), pout = 0.75 (out pressure) and U = 0.05 (rotor velocity), the Mach number values range theinterval [0.025, 0.41]Fig. 2. An illustration of the time sequence of the stator-rotor geometric configurations. For n = 0, themeshes of the two subdomains are joint at one node, while at n = N the meshes share all nodes on thediscretized stator-rotor interface. Obviously, the configuration n = 0 can be released due to the assumedquasi-T -periodicity of the state problem solutions.channels, respectively, each embedding a single turbine blade profile, as shown in Fig. 2. Thesesubdomains have a common interface on which the coupling conditions are prescribed accord-ing to the actual instantaneous position of the rotor. The viscous compressible fluid obeys theNavier-Stokes equations which can be expressed in a compact dimensionless form, as follows∂w∂t+∇ · F = 0 , i.e. ∂wj∂t+∑i∂Fji∂xi= 0 , (1)w =ρρv1ρv2ρe , fEi =ρviρv1vi + pδ1iρv2vi + pδ2ivi(ρe+ p) , fVi = 1Re0τ1iτ2i∑(τijvj)− γPr∇(e− v22) ,where fEi and fVi , are the columns of FE and FV , respectively, and F = FE(w) − FV (w,∇w).The state variable w involves the density ρ, velocity (v1, v2) and the internal energy e. Theviscous part of the stress is denoted by τij . We consider laminar subsonic flows and the boundaryconditions are classically defined by the flow direction at the inlet (the angle of attack α) anda pressure at the outlet pout, while the value 1 is assigned to the inlet stagnation density andpressure. The initial conditions can be chosen almost arbitrarily, since the state problem solutionis defined as a steadily periodic solution. Although an exact periodic solution is not enforced124and, thus not guaranteed, we assume the existence of time t0 such that the solutions w(t) of theinitial value problem satisfy an approximate periodicity condition‖w(t)− w(t+ T )| <  , for all t ≥ t0 , (2)where  is a given precision and the norm ‖w‖ is represented by the Euclidean norm of thespace-discretized state vector w. Adhering to the assumption (2), we consider only the timesteps n ∈ [1, N ] such that tn = t0 + n∆t. Thus, the quasi-T -periodic solutions of the stateproblem are represented by the N -tuples {wn}Nn=1, whereby w0 ≈ wN in the sense of (2)To capture the motion of the rotor part domain, we consider a decomposition of the veloc-ity field v involved in (1) into the mesh velocity vmsh , and the relative velocity vrel, so thatv = vrel + vmsh. For the stator-rotor system, vmsh is defined piecewise constant; while it van-ishes in the stator subdomain, in the rotor subdomain, vmsh equals a constant vector U whichdescribes the revolution speed U = |U| of the rotor. Therefore, U is involved in the convectiveacceleration terms of the flow equations in the rotor domain only. The discontinuity of vmshon the stator-rotor interface requires updating the position of the mesh nodes of the rotor mesh.Consequently, pairs of the homologous nodes on the interface commute with subsequent timelevels. One time period T is subdivided into N steps ∆t = Uδd according to the interface dis-cretization δd which defines the shift of the rotor mesh position with respect to the stator ones,see Fig. 2, where the sequence of N relevant geometric configurations is displayed. Thus, fora given time level, this shift provides a particular coupling scheme for the nodal values of thediscretized quantities along the matching interfaces of the subdomains.3. Shape sensitivity of an objective functionThe shape sensitivity analysis is needed to allow for using gradient based optimization methods.In this context, we consider a general objective functionalΦ =∫ t0+Tt0ϕ(w(t), t)d t , (3)where ϕ evaluates a criterion of the flow optimality at time t. For instance ϕ can express theloss of stagnation pressure relative to the inlet kinetic energy. In general, we assume Φ isbounded below over the set of all admissible solutions of the State Problem (SP) given by (1) –(2). Since the sensitivity has been derived for the fully discretized SP, we introduce the shapeOptimization Problem (OP) in terms of the discretized SP; by Wn we refer to the discretizedstate wn. The shape of the blades profiles is described the non-uniform rational basis splinefunctions involving mP control points P i. By Y we denote the vector of m = 2mP coordinatesof the control points P i; in Fig. 3, locations of P i in the reference configuration is depicted.Below we consider the set Y of all admissible blades designs restricting the shape geometrywith respect to criteria independent of the state W.We consider the general OP: Find Y∗ ∈ Y and the state W∗ = W(Y∗), such thatΦ(Y∗,W(Y∗)) ≤ Φ(Y,W(Y)) ∀Y ∈ Y , where Φ :=N∑n=1φn(Y,Wn) ,subject to: Fn(Y,Wn,Wn−1) = 0 , n = 1, . . . , N , and WN ≈ W0 ,(4)where Fn = 0 represents the space-time discretized approximation of Problem (1) – (2), andW = {Wn}Nn=1 involves the state vectors at all relevant time levels. The admissible states W(Y)are defined as an implicit function constituted by the constraint (4)2.125Fig. 3. The rotor blade profile and positions of 32 control points in the reference configuration.Within the design-state space of any (Y.W), the sensitivity analysis provides the total designgradient dtot Φ(Y,W(Y))/dY on a manifold of admissible states W(Y),dtot ΦdY=N∑n=1(∂φn∂Y+∂φn∂WnGn), (5)where the matrix Gn = ∂Wn/∂Y is the design gradient of the state Wn. To evaluate total designgradient, we pursue the Direct Differentiation (DDM) and the Adjoint System (ASM) methods,the latter one allows to avoid computing matrices Gn, however, the adjoint vectors λn mustbe solved for. In contrast with the standard transient problems, the initial condition W0 is notknown, as explained above. Instead, the time periodicity conditions must be handled. Comput-ing λn rigorously would require solving a set of coupled linear equations for all {λn}Nn=1. Inthe case of the DDM, a similar difficulty is encountered. Therefore, we propose iterative DDMand ASM algorithms. The dimension of λn is independent of the number of design variables.Consequently, when the number of the design variables is large, the ASM is preferred to theDDM, cf. [3]. Both these methods have been implemented to our application. Numerical testsand verifications have been performed.AcknowledgmentsThis research is supported by the project “Centre of research and experimental development ofreliable energy production” TE01020068 of the Technology Agency of the Czech Republic. Theauthors also thank O. Bublík for his assistance in the numerical implementation of algorithms.References[1] Bublík, O., Pecka, A., Vimmr, J., FlowPro documentation, University of West Bohemia, 2017.https://www.kme.zcu.cz/flowpro/index.php[2] Moravcová, F., Lukeš, V., Rohan, E., Surface sensitivities computation for turbine blades us-ing OpenFOAM, Proceedings of the conference Computational Mechanics 2016, Špicˇák, 2016,pp. 79-80.[3] Rohan, E., Cimrman, R., Sensitivity analysis and material identification for activated smooth mus-cle, Computer Assisted Mechanics and Engineering Sciences 9 (4) (2002) 519-541.[4] Rohan, E., Whiteman, J., Shape optimization of elasto-plastic structures and continua, ComputerMethods in Applied Mechanics and Engineering 187 (2000) 261-288.126

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Sensitivity analysis for an almost periodic unsteady flow problem; application to turbo-machinery modelling

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