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Research Papers

Influence of Acoustic Blockage on Flutter Instability in a Transonic Nozzle OPEN ACCESS

[+] Author and Article Information
Quentin Rendu

Laboratoire de Mécanique des Fluides et
d'Acoustique,
Université Claude Bernard Lyon 1,
Université de Lyon,
43 Boulevard du 11 Novembre 1918,
Villeurbanne 69100, France
e-mail: quentin.rendu@univ-lyon1.fr

Yannick Rozenberg

Laboratoire de Mécanique des Fluides et
d'Acoustique,
Université Claude Bernard Lyon 1,
Université de Lyon,
43 Boulevard du 11 Novembre 1918,
Villeurbanne 69100, France

Stéphane Aubert

Laboratoire de Mécanique des Fluides et
d'Acoustique,
École Centrale Lyon,
Université de Lyon,
36, avenue Guy de Collongue,
Ecully 69134, France
e-mail: stephane.aubert@ec-lyon.fr

Pascal Ferrand

Laboratoire de Mécanique des Fluides et
d'Acoustique,
École Centrale Lyon,
Université de Lyon,
36, avenue Guy de Collongue,
Ecully 69134, France

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received March 14, 2017; final manuscript received September 28, 2017; published online November 21, 2017. Editor: Kenneth Hall.

J. Turbomach 140(2), 021004 (Nov 21, 2017) (10 pages) Paper No: TURBO-17-1044; doi: 10.1115/1.4038279 History: Received March 14, 2017; Revised September 28, 2017

In order to predict oscillating loads on a structure, time-linearized methods are fast enough to be routinely used in design and optimization steps of a turbomachine stage. In this work, frequency-domain time-linearized Navier–Stokes computations are proposed to predict the unsteady separated flow generated by an oscillating bump in a transonic nozzle. The influence of regressive pressure waves on the aeroelastic stability is investigated. This case is representative of flutter of a compressor blade submitted to downstream stator potential effects. The influence of frequency is first investigated on a generic oscillating bump to identify the most unstable configuration. Introducing backward traveling pressure waves, it is then showed that aeroelastic stability of the system depends on the phase shift between the wave's source and the bump motion. Finally, feasibility of active control through backward traveling pressure waves is evaluated. The results show a high stabilizing effect even for low amplitude, opening new perspectives for the active control of choke flutter.

The prediction of oscillating loads on a structure is a key component of fluid–structure interaction analysis. This prediction can nowadays be achieved thanks to nonlinear unsteady Reynolds-averaged Navier–Stokes (URANS) methods. These methods are accurate enough to identify some starting instabilities, such as surge or flutter, and they can deal with complex geometries, but still at high restitution delays. Such computations are prohibitive for design or optimization purposes where a lot of configurations are examined. For periodic flows, time-linearized methods can predict the harmonic response of the flow without computing the transient regime between steady and periodic states. These methods are thus 10–100 times faster than standard nonlinear time-marching techniques.

Time-linearized methods have been widely used in turbomachinery for aeroelastic studies [14]. In time-linearized methods, the unsteady flow is described as the sum of the steady solution and a harmonic perturbation at a prescribed frequency. Linearizing the Navier–Stokes equations around a steady-state results in a system of linear equations, which can be quickly solved using iterative Krylov subspace methods [5,6]. Through linearized methods, only a single harmonic of the periodic response of the flow is obtained. This may be enough to estimate the aerodynamic damping around a vibrating blade but not necessarily sufficient for a deeper analysis of the flow, especially if more than one harmonic drives the unsteadiness.

If these methods have been widely used for inviscid or viscous attached flows, the prediction of turbulence response is a great challenge, especially for dynamic separation of boundary layers. Moreover, as pointed out by Szechenyi [7], what is called stall flutter generally occurs without stall but in the presence of strong shock wave and separated boundary layer. This is confirmed by recent studies of Aotsuka and Murooka [8] and Vahdati and Cumpsty [9]. In such transonic flows, the importance of backward traveling waves on aeroelastic stability through acoustic blockage has also been demonstrated [10,11]. Reliable aeroelastic computations modeling shock wave/boundary layer interaction and propagation of acoustic waves are thus mandatory to predict transonic flutter.

In this paper, the authors make use of experimental results obtained by Ferria [12] obtained from the KTH-VM100 facility. The experimental setup is presented in the first part. Aeroelastic nonlinear and time-linearized numerical methods are then exposed. Computations are performed to analyze the aeroelastic behavior of shock wave/boundary layer interaction in the third part. Finally, the influence of acoustic blockage on the instability is assessed.

Oscillating Bump.

The experimental setup used for validation consists in a generic model whose profile is similar to a typical compressor blade. It is located at Royal Institute of Technology (KTH) in the VM100 facility transonic nozzle (see Fig. 1). In the present work, the generic model considered is a bi-dimensional polyurethane bump, which can oscillate at different frequencies. The mode shapes obtained are expected to be similar to the first bending mode of a compressor blade. The height of the nozzle is equal to the axial chord length of the bump cax = 120 mm. A sketch of the bi-dimensional polyurethane bump is presented in Fig. 2.

The bending mode is excited thanks to a rotating camshaft. The prismatic shape of the camshaft allows an oscillation of the flexible bump up to a frequency of 500 Hz (corresponding to a driven rotation speed of 10,000 rpm) [13]. The motion of the bump is sinusoidal with expected amplitude 0.5 mm.

Steady pressure coefficients were obtained along the bump by Ferria with an accuracy of 0.04% full scale [12]. Unsteady pressure coefficients were obtained along the bump for different frequencies ranging from 10 Hz to 300 Hz, corresponding to a range of reduced frequency kr ∈ [0.005, 0.15]. The reduced frequency is based on the full axial chord cax and the velocity measured at the nozzle outlet uexit = 240 m/s Display Formula

(1)kr=fcaxuexit

As pointed out by Andrinopoulos et al. [14], the oscillation of the polyurethane bump does not reproduce a pure bending mode and the amplitude varies with frequency. However, the mode shape is known at each frequency from dynamic measurements, allowing the validation on this configuration. An aluminum bump has been designed to correct this issue but the dynamic results are still to be obtained. The experimental results used for validation in this work are thus obtained with the polyurethane bump. The deformation measured at f = 100 Hz (corresponding to a reduced frequency kr = 0.05) is used for unsteady computations. The dynamic deformation does not vary along spanwise direction, assessing the bi-dimensionality of the blade oscillation. Several dynamic measurements at different frequencies show that the experiment is highly repeatable [12].

Perturbations Generator.

As presented in Fig. 1, an acoustic perturbations generator consisting of an elliptic camshaft is located downstream of the test section in the nozzle. When rotating, it generates acoustic waves, which propagate and excite the steady shock wave. Several elliptic camshafts were designed to generate back pressure perturbations with controlled amplitude and frequency up to 500 Hz. The amplitude of these waves ranges up to 2% of the mean outlet static pressure. Bron et al. performed unsteady pressure measurements on a fixed bi-dimensional bump for different frequencies and amplitude of back pressure fluctuations [15].

Fluid/Structure Coupling.

To evaluate the stability of the fluid/structure interaction, we use in this work the energy method. It consists in computing the work done by aerodynamic forces when the structure is oscillating. Through this method, weak coupling is assumed between fluid and structure, meaning that the unsteady aerodynamic forces do not impact the structure mode shape and frequency. Turbomachines blades are stiff in comparison to airplane wings, and this approach is thus commonly used in turbomachinery applications [16].

Aerodynamic Damping Parameter.

To obtain quantitative information on aerodynamic stability, one may want to compute the aerodynamic damping parameter as formulated by Verdon [17]. This normalized parameter quantifies the energy exchanged between the fluid and the structure. If positive, the mechanical energy of the vibrating structure is extracted by the surrounding flow, leading to aerodynamic stability. If negative, energy is given by the flow to the structure, leading to aerodynamic instability. In case of aerodynamic instability, the overall stability is driven by the sum of the negative aerodynamic damping parameter and the structural damping parameter (computed from the type of embedding, the materials…). If the global damping parameter is positive in spite of an aerodynamic instability, the system succeeds in dissipating mechanically the energy extracted from the surrounding flow.

The aerodynamic damping parameter ζ is defined as Display Formula

(2)ζ=14πΩWdΩU

where Ω is the fluid–structure contact surface, W=0TP(x,t)dt=(0TP̃(x,t)dt) is the exchanged work between the flow and the structure, P̃ is the instantaneous complex exchanged power, and U is the maximal kinetic energy of the vibration. U is defined as Display Formula

(3)U=12ρsolidSLωf2Δymax2

where Δymax is the maximal displacement of the bump in the crosswise direction, ρsolidSL = 0.08 kg is the mass of the oscillating part of the bump for a spanwise dimension L = 0.1 m (S = 8.04 cm2, ρsolid = 1030 kg/m3). In the case of a pure bending mode, the exchanged power can be expressed as Display Formula

(4)P̃(x,t)=Ps̃(x,t)×Ldx×n(x)·Ṽ*(x,t)

where Ps̃ is the instantaneous complex static pressure, Ṽ is the instantaneous complex velocity vector of the structure, Ldx is the local steady surface of the bump (dx being an infinitesimal streamwise length), and n is the normal vector to this surface oriented outward of the flow. Knowing the fundamental frequency f = ωf/2π of the unsteady flow, the instantaneous variables can be written Display Formula

(5)w̃(x,t)=0w(x)+n>0w̃n(x)ej(nωf)t

where 0w(x) is the steady part of the variable w̃ and w̃n(x) is the complex amplitude of the nth harmonic. Keeping only the first harmonic (i = 1), we can write Display Formula

(6)0TP̃(x,t)dt=0TPs̃(x,t)×Ldx×n·Ṽ*(x,t)dt
Display Formula
(7)=T×1Ps̃(x)×Ldx×0n(x)·1Ṽ*(x)

and we can finally obtain the expression of the aerodynamic damping parameter Display Formula

(8)ζ=14πT(wallPs̃1(x)×0n(x)·1Ṽ*(x)dx)12ρsolidSωf2Δymax2

Time-Linearized RANS Equations.

By linearizing the Navier–Stokes equations around a steady solution, we assume that the unsteady flow can be expressed as the sum of a steady part and a harmonic perturbation. Formally, if q(ρ,ρu,ρE,ρk,ρω) is the vector of conservative and turbulent variables, the finite volume formulation of Navier–Stokes equations can be written as Display Formula

(9)ddt[J(p)q]+F(p,q)=0

where p is the vector determining boundary conditions, J(p) is the vector of mesh cells' volume, and F is a nonlinear operator representing the balance between convective and viscous fluxes. Let us assume that the vectors p and q can be written as the sum of a steady part (only space dependent) and a harmonic unsteady part Display Formula

(10)p(x,t)=0p(x)+1p̃(x)ejωft
Display Formula
(11)q(x,t)=0q(x)+1p̃(x)ejωft

where ωf is a prescribed pulse. Note that the complex amplitude of perturbation vectors p̃1 and p̃1 must be small in comparison to p and q to achieve linearization. Within this formalism, the space-dependent steady vectors 0p and 0q satisfy Display Formula

(12)F(p0,q0)=0

Finally, we can write the linearized RANS (LRANS) equations Display Formula

(13){jωfJ(p0)+Fq(p0,q0)}1q̃={jωfJp(p0)+Fp(p0,q0)}1p̃

In order to solve this equation, one must compute the derivatives of the nonlinear operator F according to the vectors p and q. To this end, the partial derivatives are exactly evaluated by applying an automatic derivation tool to the F function coming from the RANS solver Turb'Flow [18,19]. These partial derivatives are then read by the LRANS solver Turb'Lin [20] used in this work.

Concerning the turbulence modeling, two different approaches are commonly used. If the time for the turbulence to react to the unsteadiness is high in comparison to the characteristic time of the unsteadiness (typically a period of oscillation for flutter analysis), it can be assumed that the turbulence behavior is frozen to its steady-state. On the other hand, one can assume that the turbulence depends on the unsteady flow. Philit et al. [21] and Rendu et al. [22] showed that the latter assumption, called harmonic turbulence linearized RANS, is mandatory to reproduce the unsteady behavior of shock wave/boundary layer interaction. Thus, the harmonic turbulence assumption is used in this work. The turbulence model chosen for the computations is the classical kω model of Wilcox [23]. The underlying assumptions of such a model are turbulence isotropy and immediate response of the turbulence to the flow. These assumptions, specific to the turbulence model and not to the time linearization, may not be well suited for nonequilibrium flows such as stall and shock wave/boundary layer interaction. Nevertheless, this study aims at comparing LRANS and URANS computations, even if all physical mechanisms are not perfectly captured.

Numerical Solver.

The solver Turb'Flow [18,19], developed at Laboratory of Fluid Mechanics and Acoustics in Ecole Centrale de Lyon, is used for steady RANS and unsteady RANS computations. Turb'Flow solves the compressible three-dimensional (3D) Navier–Stokes equations using finite volume formulation. Centered and upwind spatial scheme are implemented, as well as several turbulence models. Both explicit and implicit methods are available for temporal discretization to achieve faster computations. In this work, spatial discretization of convective fluxes is achieved by an upwind Roe scheme with third-order MUSCL interpolation and harmonic cubic upwind interpolation (HCUI) flux limiter on conservative variables and with a second-order MUSCL interpolation with sharp monotonic algorithm for realistic transport equation revisited (SMARTER) flux limiter on turbulent variables. Center finite difference is used to discretize the viscous fluxes of both conservative and turbulent variables. The temporal discretization for URANS computations is achieved thanks to a dual time-stepping method.

Unsteady Boundary Conditions
Bump Oscillation.

For nonlinear unsteady RANS computations of the oscillating bump, the mesh must be deformed during the computation. As the mesh cells are thinner at the wall, a rigid body motion is imposed to the cells in the boundary layer. The mesh deformation is achieved at the center of the domain, where the cells are larger, as shown in Fig. 3. With this technique, the volume variation of the most deformed cell is lower than 1%, guaranteeing almost the same quality than the steady mesh.

The numerical imposed deformation is computed thanks to the results of Andrinopoulos et al. [14] on the polyurethane bump Display Formula

(14)Δy=a0a2(xx0)2+a3(xx0)3

The parameters in Eq. (14) are given in Table 1. The amplitude of the numerical deformation compared to experimental data is plotted along the bump in Fig. 4 as well as the steady and extremes numerical positions of the bump in Fig. 5. The origin of the abscissa axis is chosen 70 mm upstream of the bump.

Backward Travelling Pressure Waves.

The backward traveling pressure waves generated experimentally by an elliptic camshaft are modeled through the fluctuation of static pressure at the domain outlet. Both amplitude and phase of the fluctuating pressure are imposed.

Pressure Coefficients.

To compare experimental and numerical results, the steady pressure coefficient is computed for steady-state as Display Formula

(15)Cp=PsPtinletPsinletCpinlet

where Cpinlet is defined as Display Formula

(16)Cpinlet=PsinletPtinletPsinlet

Note that to take into account the change in outlet static pressure in two-dimensional (2D) computations, the experimental and numerical values of Cpinlet are different. For unsteady calculations, the unsteady pressure coefficient is computed, consistently with Ferria formulation [12] as Display Formula

(17)Cp̃1=Ps̃1PtinletPsinlet×caxA

where Ps̃1 is the complex amplitude of the first harmonic of the instantaneous static pressure, cax = 120 mm is the axial chord length, and A = 0.5 mm is the theoretical amplitude of the bump motion.

Steady-State Results.

The experimental setup is composed of two converging ducts upstream of the bump, which are not computed in this work. Philit [24] took into account these ducts in 3D computations to obtain an inlet boundary-layer profile suitable for 2D studies, as well as the corresponding mesh. In this study, we use the 2D structured 209 × 84 H mesh of Philit, presented in Fig. 6.

Temporal discretization is achieved through implicit method with smoothed local time-step (10% maximum variation between adjacent grid points) and maximum CFL of 50. Turbulence modeling is achieved through Wilcox (1988) kω [23]. The resulting system of linear equations is solved with a GMRES Krylov subspace method [25]. In the studied configuration, inlet total pressure of 160 kPa is imposed. The experimental results were obtained with an outlet static pressure of 104 kPa, corresponding to a strong shock wave with a separated boundary layer. To reproduce this phenomenon in 2D computations, the outlet static pressure was set to 108 kPa, as proposed by Labit et al. [26]. Steady contours of Mach number are plotted in Fig. 7 in the bump vicinity. The transonic nozzle is choked, a strong shock wave impacts the bump at the steady separation point S (xS = 145 mm) and the boundary layer reattaching at steady reattachment point R (xR = 201 mm).

The steady pressure coefficient obtained with kω turbulence model is compared in Fig. 8 with experimental results of Ferria [12]. The experimental extreme positions of the shock-wave pressure rise obtained by Bron [27] are also presented in the figure.

The overall agreement between experimental and numerical data is good. The position of the shock wave is well predicted but the pressure recovery is slightly overestimated in the separated boundary layer.

Unsteady Results.

Philit et al. [21] computed the unsteady loads generated by acoustic perturbations on a steady bump through nonlinear and time-linearized methods. These results were successfully compared to experimental data of Bron et al. [15] in order to assess the numerical methods. Hence, this work focuses on the prediction of the unsteady loads on the oscillating bump.

The unsteady experimental results of Ferria [12] at frequency f = 100 Hz, corresponding to a reduced frequency kr = 0.05, are chosen to assess the unsteady numerical results. Previous work showed that 64 dual-time stepping time steps are enough to achieve temporal convergence in URANS computations [22]. Moreover, the first harmonic contains more than 99% of the unsteady signal power [22]. Hence, in this work, only the first harmonic is used to compute unsteady pressure coefficient and local aerodynamic work in URANS computations.

The amplitude of the first harmonic of the unsteady pressure coefficient is presented in Fig. 9 for both numerical and experimental results. Experimental data show three zones of high amplitude: (i) upstream of the bump, (ii) in the vicinity of the strong shock wave, and (iii) at the reattachment of the separated boundary layer. Low values of amplitude are experimentally found in the supersonic zone (i.e., at the top of the bump, where the deformation of the bump is maximum) and downstream of the shock-wave root.

Unsteady Reynolds-averaged Navier–Stokes computation predicts accurate amplitudes from inlet to the shock wave. The drop of amplitude downstream of the shock wave and the fluctuating pressure in separated boundary layer are underestimated. Moreover, the position of maximum amplitude at the reattachment point of the boundary layer is not perfectly captured.

The amplitude of the pressure coefficient only represents the aerodynamic unsteady loads applied on the structure over one period of oscillation. The phase of the pressure coefficient, representing the temporal phase shift between pressure fluctuations and wall velocity, drives the stability of the system. Thus, the prediction of the phase difference between pressure and imposed wall velocities is the main challenge in fluid–structure interaction analysis. In Fig. 9 is plotted the phase of the first harmonic of the unsteady pressure coefficient. The configuration studied by Ferria at kr = 0.05 (f = 100 Hz) is stable (Φ ≥ −180 deg), what is correctly predicted by URANS computations. These show good agreement upstream of the bump. However, there is a 30–50 deg phase shift between the experimental results and the URANS results in the separated boundary layer (145 mm < x < 201 mm). Concerning linearized results, LRANS results presented in Fig. 9 show great ability in reproducing both amplitude and phase obtained by nonlinear method.

The mechanisms of the shock wave/boundary layer interaction are, at first glance, well estimated by Wilcox (1988) kω turbulence model. More advanced turbulence closures, such as Wilcox (2006) kω [28] or Reynolds stress model [29], show even better agreement on this configuration [22]. It is to keep in mind that the experimental flow is three-dimensional and exhibits large corner separations downstream of the shock wave. Thus, whether the discrepancies observed in this work are a consequence of turbulence modeling and/or three-dimensional effects is still unclear. If three-dimensional effects are important, 3D LRANS computations allow to evaluate their influence on aeroelastic analysis. To investigate the sensibility to turbulence modeling, a regularized (derivable) formulation of the turbulence model must be implemented, which may require some effort. A methodology of regularization, applied to the Wilcox (2006) two-equation model, has been proposed by Rendu et al. [30].

Finally, the ability for linearized method to reproduce nonlinear results is demonstrated here, even in presence of shock wave/boundary layer interaction. To this end, the harmonic turbulence assumption must be used, what implies from Eq. (13) to fully derive the turbulence model (two equations in case of two-equation turbulence model).

Aeroelastic Stability Analysis.

If the phase shift between pressure fluctuations and imposed wall velocities gives qualitative insights in the aerodynamic stability of the system, it is generally not sufficient to conclude on the global stability. To go further, exchanged work W is plotted in Fig. 10 along the bump for URANS and LRANS computations. Notice that when the exchanged work is positive, the energy of the structure is dissipated by the surrounding fluid, leading to aerodynamic stability. A stabilizing peak is observed in the vicinity of the shock wave, where pressure fluctuations are strong, and the contribution of the separated boundary layer downstream of the shock wave is also important. The damping parameter computed from nonlinear results is ζURANS = 0.1. The relative error obtained using linearized method is equal to 0.2%. As the damping parameter is very well predicted by LRANS computations, we can now take advantage of linearized method to study the aerodynamic stability over a range of frequencies.

The aerodynamic damping parameter ζ is presented in Fig. 11 for 0.1 < kr < 0.4 corresponding to a range of frequency 200 Hz < f < 800 Hz. Up to kr = 0.15, the damping parameter gradually decreases and reaches a minimum value ζ = −0.7 × 10−3. Then, the damping parameter increases and aerodynamic stability is recovered for kr = 0.25 and above. Through linearized Euler analysis of airfoil cascades, Corral and Vega showed that, for small reduced frequency, the work-per-cycle scales linearly with the reduced frequency [31]. Here, we find a nonlinear behavior of the damping parameter with respect to frequency, around kr = 0.15. It shows the great influence of the turbulence on the unsteady response of the flow in our configuration. It is also to notice that the modal shape has been kept identical, whereas it slightly varies with the frequency when using the polyurethane bump, as detailed in Sec. 2.1. Nevertheless, this methodology allows us to find the frequency at which the system is the most unstable. We can now run a full nonlinear unsteady RANS computation to further investigate the physical behavior of the system at kr = 0.15 (f = 300 Hz).

In Fig. 12 is plotted the exchanged work W along the bump for this frequency. The main difference between the stable and the unstable configurations is the role of the shock wave. In both cases, the unsteady loads generated by the shock oscillation are very important but the work extracted is stabilizing at kr = 0.05 and destabilizing at kr = 0.15.

The ability of LRANS computations to predict pressure fluctuations in shock wave/boundary layer interaction has been demonstrated. In Sec. 5, LRANS method is used to investigate the influence of acoustic blockage on the aeroelastic response.

In this part, back pressure perturbations are generated to interact with the oscillating shock wave. The amplitude of these perturbations and the phase shift between bump motion and acoustic waves are imposed. Through linearized theory, we expect that the unsteady flow will be the sum of the responses to the two sources of unsteadiness: bump motion and acoustic perturbations. To validate this hypothesis, nonlinear computations are run with a significant value of perturbations amplitude (±2.12 × 103 Pa, corresponding to ±2% of the outlet static pressure).

In Fig. 13 are plotted the amplitude and phase of the first harmonic of unsteady pressure coefficient from nonlinear URANS and LRANS computations. In this configuration, both pressure waves and bump motion contribute to the unsteady flow. The phase shift between both excitations is ΔΦ = 0 deg. There is a very good agreement between URANS and LRANS results, showing that the unsteady behavior is linear for this range of amplitude (δPs/Psoutlet = 2%). The same variables are plotted in Fig. 14 for a configuration with a phase shift between both excitations of ΔΦ = 90 deg. Nonlinear and linear predictions are almost identical, demonstrating that linearity is achieved for different phase shifts.

Thanks to linearized method, the unsteady response of the flow to different sources of unsteadiness can be decomposed and separately analyzed. In Fig. 15 (resp. Fig. 16) is plotted the amplitude (resp. the phase) of first harmonic of unsteady pressure for the two types of excitation: bump motion (top) and back-pressure fluctuations (bottom).

Looking at the flow generated by the bump motion, one can notice the propagation of a plane wave upstream of the bump. This is particularly clear on the phase field (Fig. 16(a)) on which the phase varies linearly with the distance to the bump. In the vicinity of the bump, huge pressure fluctuations are naturally spotted not only at the shock-wave root but also in the separated boundary layer, emphasizing the role of turbulence on the aeroelastic response. Further downstream, the amplitude decreases with the distance to the bump, indicating that the waves generated vanish, which corresponds to an acoustic cut-off mode.

Concerning the unsteady response of the flow due to back pressure fluctuations, no perturbation is seen upstream of the shock wave (amplitude is null, see Fig. 15(b)). Indeed, acoustic waves are reflected by the shock and no information propagates through it. Thus, computing the phase ahead of the shock does not make sense. At the domain outlet, a pressure fluctuation δPs/Psoutlet = 0.25% is imposed, which propagates as a plane wave in the duct (see Fig. 16(b)). The amplitude decreases and even vanishes close to the lower wall while it increases ahead of the shock wave in the upper part of the duct, where sonic conditions prevail (see Fig. 7). This phenomenon, known as acoustic blockage, was first described by Atassi et al. [11].

To analyze the aeroelastic behavior of the oscillating bump, the extracted work is plotted along the bump in Fig. 17. The solid line corresponds to the work extracted by the oscillating bump without back-pressure fluctuations, whereas the dashed lines represent the work with back-pressure fluctuations with different phase shifts ΔΦ = 0 deg and ΔΦ = 90 deg corresponding to Figs. 13 and 14, respectively. As expected, the acoustic waves have no influence on the work extracted upstream of the shock wave (see Fig. 15(b)). These perturbations impact the behavior of the shock wave and the separated boundary layer either by stabilizing this zone or destabilizing it, depending on the value of ΔΦ. Following this idea, the influence of ΔΦ on the total damping parameter is investigated in the next paragraph.

The observed (total) aerodynamic damping parameter of the oscillating bump ζ can be decomposed in two parts Display Formula

(18)ζ=ζ0+Δζ

where ζ0 corresponds to the damping parameter of the oscillating bump without acoustic perturbations (ζ0 = −0.7 × 10−3 at f = 300 Hz from Fig. 11), whereas Δζ is the added damping parameter generated by the acoustic perturbations. Δζ depends on both amplitude and phase shift of back-pressure fluctuations. Δζ is plotted in Fig. 18 as a function of the phase shift for two different amplitudes. These results are obtained from linearized computations and validated thanks to right nonlinear URANS computations. From the figure, one can observed that the added damping parameter is close to a sine function of variable ΔΦ at a given frequency. This can be generalized as Display Formula

(19)Δζ=Z(f)×A×sin(θ(f)+ΔΦ)

where A is the amplitude of back-pressure fluctuations (in %), Z(f) the maximal added damping parameter for a back-pressure fluctuations of 1% and θ(f) an arbitrary phase shift. In our configuration, we find θ(300) = −41 deg and Z(300) = 0.0073. Hence, by generating acoustic perturbations of amplitude 2%, one can expect a maximal added damping parameter of 0.0146 for ΔΦ = 131 deg (see Fig. 18).

In Sec. 4.3, we found a minimal aeroelastic stability for kr = 0.15 (f = 300 Hz). It corresponds to an aerodynamically unstable system with a damping parameter ζ0 = −0.7 × 10−3. The feasibility of control through single frequency pressure waves is now briefly investigated. To effectively recover aerodynamic stability, one wants to maximize the added damping parameter for a given amount of energy. To this end, acoustic perturbations should be generated with a phase shift ΔΦ = 131 deg. We choose an amplitude of 0.25% (corresponding to 265 Pa), what is motivated by a very low amount of energy needed as well as a sufficient margin to recover stability (ζ = ζ0 + Δζ = 1.1 × 10−3).

As the device consists of an elliptic camshaft located downstream of the oscillating bump, the amplitude of the back-pressure fluctuations is driven by the geometry. However, the phase shift ΔΦ between the bump motion and the rotating camshaft has to be controlled. To this end, we propose to record the unsteady pressure on the bump at two different points xC = 115 mm and xR = 201 mm (see Fig. 7) for different phase shifts ΔΦ. In Fig. 19 are plotted, along ΔΦ, the added damping parameter for an amplitude of 0.25% and the phase observed at these two points. The use of two probes ensures a wider operating range and prevents measurements in a region of too low amplitude. Moreover, such probes can be used to detect a starting instability and trigger the rotation of the elliptic camshaft. The maximal added damping parameter is obtained for ΔΦ = 131 deg corresponding to Φ(xR) = 45 deg and Φ(xC) = 22.5 deg. Thus, a mechanical device driving the camshaft phase shift linked to two kulite transducers on the bump with a feedback control loop can be used as a first step to achieve control of the aerodynamic instability.

In this work, the ability of time-linearized method to reproduce nonlinear RANS simulations has been demonstrated in case of shock wave/boundary layer interaction over an oscillating bump. This result was achieved by taking into account the unsteady behavior of the turbulence through harmonic turbulence assumption. Time-linearized computations have then been used to evaluate the influence of frequency on the aeroelastic stability to identify the most unstable configuration.

Finally, we evaluated the influence of back-pressure fluctuations on the aeroelastic behavior of the model. These perturbations may excite or damp the oscillating system, depending on the phase shift between acoustic waves and system vibration. This opens new perspective for the control of flutter instabilities.

The promising results, obtained with 2D computations, should be extended to three-dimensional flows to investigate the impact of corner separations on the unsteady response of the flow. Experimental campaign in the KTH-VM100 facility could also confirm the effect of acoustic blockage and use it to control transonic flutter.

The methodology proposed in this work to identify the unstable configurations may already be applied in production environment. Adding backward pressure fluctuations, it can also be used in research departments to investigate the design of active control devices of transonic flutter.

  • A =

    amplitude of bump oscillation (m)

  • cax =

    axial chord (m)

  • Cp =

    steady pressure coefficient

  • f =

    frequency (Hz)

  • j =

    complex number j2 = −1

  • kr =

    reduced frequency

  • L =

    spanwise dimension of the bump (m)

  • M =

    Mach number

  • P =

    extracted power (W)

  • Ps =

    static pressure (Pa)

  • Pt =

    total pressure (Pa)

  • (z̃) =

    real part of complex number z̃

  • t =

    time (s)

  • T =

    time period of one oscillation (s)

  • Ts =

    static temperature (K)

  • u =

    fluid velocity vector (m s−1)

  • U =

    maximal kinetic energy of vibration (J)

  • x =

    position in streamwise direction (m)

  • W =

    extracted work (J)

  • w̃n(x) =

    complex amplitude of nth harmonic of instantaneous complex variable w̃(x,t)

  • 0w(x) =

    steady part of instantaneous complex variable w̃(x,t)

  • z̃* =

    conjugate of complex number z̃

  • |z̃| =

    amplitude of complex number z̃

  • Δy =

    displacement of the bump (m)

  • ζ =

    damping parameter

  • ρ =

    density (kg m−3)

  • Φ(z̃) =

    phase of complex number z̃

  • ωf =

    pulse (ωf = 2πf) (s−1)

Clark, W. S. , and Hall, K. C. , 2000, “ A Time-Linearized Navier–Stokes Analysis of Stall Flutter,” ASME J. Turbomach., 122(3), pp. 467–476. [CrossRef]
Campobasso, M. , and Giles, M. , 2003, “ Effects of Flow Instabilities on the Linear Analysis of Turbomachinery Aeroelasticity,” J. Propul. Power, 19(2), pp. 250–259.
Kersken, H.-P. , Frey, C. , Voigt, C. , and Ashcroft, G. , 2012, “ Time-Linearized and Time-Accurate 3D RANS Methods for Aeroelastic Analysis in Turbomachinery,” ASME J. Turbomach., 134(5), p. 051024. [CrossRef]
Chassaing, J.-C. , and Gerolymos, G. , 2008, “ Time-Linearized Time-Harmonic 3D Navier-Stokes Shock-Capturing Schemes,” Int. J. Numer. Methods Fluids, 56(3), pp. 297–303.
Ipsen, I. C. , and Meyer, C. D. , 1998, “ The Idea Behind Krylov Methods,” Am. Math. Mon., 105(10), pp. 889–899. [CrossRef]
Chassaing, J. , Gerolymos, G. , and Jeremiasz, J. , 2006, “GMRES Solution of Compressible Linearized Navier-Stokes Equations Without Pseudo-Time-Marching,” AIAA Paper No. 2006-688.
Szechenyi, E. , 1987, “ Understanding Fan Blade Flutter Through Linear Cascade Aeroelastic Testing,” Aeroelasticity in Axial-Flow Turbomachines, Vol. 1, M. F. Platzer , and F. O. Carta , eds., Defense Technical Information Center, Fort Belvoir, VA, AGARD Manual AD-A181 646.
Aotsuka, M. , and Murooka, T. , 2014, “Numerical Analysis of Fan Transonic Stall Flutter,” ASME Paper No. GT2014-26703.
Vahdati, M. , and Cumpsty, N. , 2016, “ Aeroelastic Instability in Transonic Fans,” ASME J. Eng. Gas Turbines Power, 138(2), p. 022604. [CrossRef]
Ferrand, P. , 1987, “Parametric Study of Choke Flutter With a Linear Theory,” Advanced Technology for Aero Gas Turbine Components, North Atlantic Treaty Organization, Brussels, Belgium, AGARD-CP-421.
Atassi, H. , Fang, J. , and Ferrand, P. , 1995, “Acoustic Blockage Effects in Unsteady Transonic Nozzle and Cascade Flows,” AIAA Paper No. 95-0303.
Ferria, H. , 2007, “Experimental Campaign on a Generic Model for Fluid Structure Interaction Studies,” Master's thesis, KTH Royal Institute of Technology, Stockholm, Sweden. https://www.diva-portal.org/smash/get/diva2:459036/FULLTEXT01.pdf
Allegret-Bourdon, D. , 2004, “Experimental Study of Fluid-Structure Interactions on a Generic Model,” Licentiate thesis, KTH Royal Institute of Technology, Stockholm, Sweden. https://www.diva-portal.org/smash/get/diva2:7787/FULLTEXT01.pdf
Andrinopoulos, N. , Vogt, D. , Hu, J. , and Fransson, T. , 2008, “Design and Testing of a Vibrating Test Object for Investigating Fluid-Structure Interaction,” ASME Paper No. GT2008-50740.
Bron, O. , Ferrand, P. , and Fransson, T. H. , 2006, “ Experimental and Numerical Study of Non-Linear Interactions in Two Dimensional Transonic Nozzle Flow,” 11th International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Moscow, Russia, Sept. 4–8, pp. 463–481.
Marshall, J. , and Imregun, M. , 1996, “ A Review of Aeroelasticity Methods With Emphasis on Turbomachinery Applications,” J. Fluids Struct., 10(3), pp. 237–267. [CrossRef]
Verdon, J. , 1987, “ Linearized Unsteady Aerodynamic Theory,” Aeroelasticity in Axial-Flow Turbomachines, Vol. 1, M. F. Platzer , and F. O. Carta , eds., North Atlantic Treaty Organization, Brussels, Belgium, AGARD Manual AD-A181 646.
Smati, L. , Aubert, S. , and Ferrand, P. , 1996, “ Numerical Study of Unsteady Shock Motion to Understand Transonic Flutter,” 349th EUROMECH-Colloquium: Simulation of Fluid-Structure Interaction in Aeronautics, Gottingen, Germany, Sept. 16–19.
Smati, L. , Aubert, S. , Ferrand, P. , and Masao, F. , 1997, “ Comparison of Numerical Schemes to Investigate Blade Flutter,” Eighth International Symposium of Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Stockholm, Sweden, Sept. 14–18, pp. 749–763.
Philit, M. , Blanc, L. , Aubert, S. , Lolo, W. , Ferrand, P. , and Thouverez, F. , 2011, “ Frequency Parametrization to Numerically Predict Flutter in Turbomachinery,” Fourth International Conference on Computational Methods for Coupled Problems in Science and Engineering, Kos Island, Greece, June 20–22.
Philit, M. , Ferrand, P. , Labit, S. , Chassaing, J.-C. , Aubert, S. , and Fransson, T. , 2012, “ Derivated Turbulence Model to Predict Harmonic Loads in Transonic Separated Flows Over a Bump,” 28th International Congress of Aeronautical Sciences (ICAS), Brisbane, Australia, Sept. 23–28, pp. 2713–2723. http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A647189&dswid=-9940
Rendu, Q. , Philit, M. , Labit, S. , Chassaing, J.-C. , Rozenberg, Y. , Aubert, S. , and Ferrand, P. , 2015, “ Time-Linearized and Harmonic Balance Navier-Stokes Computations of a Transonic Flow Over an Oscillating Bump,” 14th International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Stockholm, Sweden, Sept. 8–11, Paper No. S11-5. https://www.researchgate.net/publication/291820780_Time-Linearized_and_Harmonic_Balance_Navier-Stokes_Computations_of_a_Transonic_Flow_over_an_Oscillating_Bump
Wilcox, D. C. , 1988, “ Reassessment of the Scale Determining Equation for Advanced Turbulence Models,” AIAA J., 26(11) , pp. 1299–1310.
Philit, M. , 2013, “ Modélisation, simulation et analyse des instationnarités en écoulement transsonique décollé en vue d'application a l'aéroélasticité des turbomachines,” Ph.D. thesis, Ecole Centrale de Lyon, Lyon, France.
Saad, Y. , and Schultz, M. , 1986, “ GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems,” SIAM J. Sci. Stat. Comput., 7(3), pp. 856–869. http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=D6B6680EE4209988F125B700624D0D82?doi=10.1.1.476.951&rep=rep1&type=pdf
Labit, S. , Chassaing, J.-C. , Philit, M. , Aubert, S. , and Ferrand, P. , 2016, “ Time-Harmonic Navier–Stokes Computations of Forced Shock-Wave Oscillations in a Transonic Nozzle,” Comput. Fluids, 127, pp. 102–114.
Bron, O. , 2004, “ Numerical and Experimental Study of the Shock-Boundary Layer Interaction in Unsteady Transonic Flow,” Ph.D. thesis, Ecole Centrale de Lyon, Lyon, France. http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A9523&dswid=8352
Wilcox, D. C. , 2008, “ Formulation of the kω Turbulence Model Revisited,” AIAA J., 46(11), pp. 2823–2838.
Gerolymos, G. , and Vallet, I. , 1997, “ Near-Wall Reynolds-Stress Three-Dimensional Transonic Flows Computation,” AIAA J., 35(2), pp. 228–236.
Rendu, Q. , Philit, M. , Rozenberg, Y. , Aubert, S. , and Ferrand, P. , 2015, “ Regularization of Wilcox (2006) kω Turbulence Model and Validation on Shock-Wave/Boundary Layer Interaction,” 12th International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows (ISAIF), Genova, Italy, July 13–16, Paper No. 119. https://www.researchgate.net/publication/283576585_Regularization_of_Wilcox_2006_k-o_turbulence_model_and_validation_on_shock_waveboundary_layer_interaction
Corral, R. , and Vega, A. , 2016, “ The Low Reduced Frequency Limit of Vibrating Airfoils—Part I: Theoretical Analysis,” ASME J. Turbomach., 138(2), p. 021004. [CrossRef]
Copyright © 2018 by ASME
View article in PDF format.

References

Clark, W. S. , and Hall, K. C. , 2000, “ A Time-Linearized Navier–Stokes Analysis of Stall Flutter,” ASME J. Turbomach., 122(3), pp. 467–476. [CrossRef]
Campobasso, M. , and Giles, M. , 2003, “ Effects of Flow Instabilities on the Linear Analysis of Turbomachinery Aeroelasticity,” J. Propul. Power, 19(2), pp. 250–259.
Kersken, H.-P. , Frey, C. , Voigt, C. , and Ashcroft, G. , 2012, “ Time-Linearized and Time-Accurate 3D RANS Methods for Aeroelastic Analysis in Turbomachinery,” ASME J. Turbomach., 134(5), p. 051024. [CrossRef]
Chassaing, J.-C. , and Gerolymos, G. , 2008, “ Time-Linearized Time-Harmonic 3D Navier-Stokes Shock-Capturing Schemes,” Int. J. Numer. Methods Fluids, 56(3), pp. 297–303.
Ipsen, I. C. , and Meyer, C. D. , 1998, “ The Idea Behind Krylov Methods,” Am. Math. Mon., 105(10), pp. 889–899. [CrossRef]
Chassaing, J. , Gerolymos, G. , and Jeremiasz, J. , 2006, “GMRES Solution of Compressible Linearized Navier-Stokes Equations Without Pseudo-Time-Marching,” AIAA Paper No. 2006-688.
Szechenyi, E. , 1987, “ Understanding Fan Blade Flutter Through Linear Cascade Aeroelastic Testing,” Aeroelasticity in Axial-Flow Turbomachines, Vol. 1, M. F. Platzer , and F. O. Carta , eds., Defense Technical Information Center, Fort Belvoir, VA, AGARD Manual AD-A181 646.
Aotsuka, M. , and Murooka, T. , 2014, “Numerical Analysis of Fan Transonic Stall Flutter,” ASME Paper No. GT2014-26703.
Vahdati, M. , and Cumpsty, N. , 2016, “ Aeroelastic Instability in Transonic Fans,” ASME J. Eng. Gas Turbines Power, 138(2), p. 022604. [CrossRef]
Ferrand, P. , 1987, “Parametric Study of Choke Flutter With a Linear Theory,” Advanced Technology for Aero Gas Turbine Components, North Atlantic Treaty Organization, Brussels, Belgium, AGARD-CP-421.
Atassi, H. , Fang, J. , and Ferrand, P. , 1995, “Acoustic Blockage Effects in Unsteady Transonic Nozzle and Cascade Flows,” AIAA Paper No. 95-0303.
Ferria, H. , 2007, “Experimental Campaign on a Generic Model for Fluid Structure Interaction Studies,” Master's thesis, KTH Royal Institute of Technology, Stockholm, Sweden. https://www.diva-portal.org/smash/get/diva2:459036/FULLTEXT01.pdf
Allegret-Bourdon, D. , 2004, “Experimental Study of Fluid-Structure Interactions on a Generic Model,” Licentiate thesis, KTH Royal Institute of Technology, Stockholm, Sweden. https://www.diva-portal.org/smash/get/diva2:7787/FULLTEXT01.pdf
Andrinopoulos, N. , Vogt, D. , Hu, J. , and Fransson, T. , 2008, “Design and Testing of a Vibrating Test Object for Investigating Fluid-Structure Interaction,” ASME Paper No. GT2008-50740.
Bron, O. , Ferrand, P. , and Fransson, T. H. , 2006, “ Experimental and Numerical Study of Non-Linear Interactions in Two Dimensional Transonic Nozzle Flow,” 11th International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Moscow, Russia, Sept. 4–8, pp. 463–481.
Marshall, J. , and Imregun, M. , 1996, “ A Review of Aeroelasticity Methods With Emphasis on Turbomachinery Applications,” J. Fluids Struct., 10(3), pp. 237–267. [CrossRef]
Verdon, J. , 1987, “ Linearized Unsteady Aerodynamic Theory,” Aeroelasticity in Axial-Flow Turbomachines, Vol. 1, M. F. Platzer , and F. O. Carta , eds., North Atlantic Treaty Organization, Brussels, Belgium, AGARD Manual AD-A181 646.
Smati, L. , Aubert, S. , and Ferrand, P. , 1996, “ Numerical Study of Unsteady Shock Motion to Understand Transonic Flutter,” 349th EUROMECH-Colloquium: Simulation of Fluid-Structure Interaction in Aeronautics, Gottingen, Germany, Sept. 16–19.
Smati, L. , Aubert, S. , Ferrand, P. , and Masao, F. , 1997, “ Comparison of Numerical Schemes to Investigate Blade Flutter,” Eighth International Symposium of Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Stockholm, Sweden, Sept. 14–18, pp. 749–763.
Philit, M. , Blanc, L. , Aubert, S. , Lolo, W. , Ferrand, P. , and Thouverez, F. , 2011, “ Frequency Parametrization to Numerically Predict Flutter in Turbomachinery,” Fourth International Conference on Computational Methods for Coupled Problems in Science and Engineering, Kos Island, Greece, June 20–22.
Philit, M. , Ferrand, P. , Labit, S. , Chassaing, J.-C. , Aubert, S. , and Fransson, T. , 2012, “ Derivated Turbulence Model to Predict Harmonic Loads in Transonic Separated Flows Over a Bump,” 28th International Congress of Aeronautical Sciences (ICAS), Brisbane, Australia, Sept. 23–28, pp. 2713–2723. http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A647189&dswid=-9940
Rendu, Q. , Philit, M. , Labit, S. , Chassaing, J.-C. , Rozenberg, Y. , Aubert, S. , and Ferrand, P. , 2015, “ Time-Linearized and Harmonic Balance Navier-Stokes Computations of a Transonic Flow Over an Oscillating Bump,” 14th International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Stockholm, Sweden, Sept. 8–11, Paper No. S11-5. https://www.researchgate.net/publication/291820780_Time-Linearized_and_Harmonic_Balance_Navier-Stokes_Computations_of_a_Transonic_Flow_over_an_Oscillating_Bump
Wilcox, D. C. , 1988, “ Reassessment of the Scale Determining Equation for Advanced Turbulence Models,” AIAA J., 26(11) , pp. 1299–1310.
Philit, M. , 2013, “ Modélisation, simulation et analyse des instationnarités en écoulement transsonique décollé en vue d'application a l'aéroélasticité des turbomachines,” Ph.D. thesis, Ecole Centrale de Lyon, Lyon, France.
Saad, Y. , and Schultz, M. , 1986, “ GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems,” SIAM J. Sci. Stat. Comput., 7(3), pp. 856–869. http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=D6B6680EE4209988F125B700624D0D82?doi=10.1.1.476.951&rep=rep1&type=pdf
Labit, S. , Chassaing, J.-C. , Philit, M. , Aubert, S. , and Ferrand, P. , 2016, “ Time-Harmonic Navier–Stokes Computations of Forced Shock-Wave Oscillations in a Transonic Nozzle,” Comput. Fluids, 127, pp. 102–114.
Bron, O. , 2004, “ Numerical and Experimental Study of the Shock-Boundary Layer Interaction in Unsteady Transonic Flow,” Ph.D. thesis, Ecole Centrale de Lyon, Lyon, France. http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A9523&dswid=8352
Wilcox, D. C. , 2008, “ Formulation of the kω Turbulence Model Revisited,” AIAA J., 46(11), pp. 2823–2838.
Gerolymos, G. , and Vallet, I. , 1997, “ Near-Wall Reynolds-Stress Three-Dimensional Transonic Flows Computation,” AIAA J., 35(2), pp. 228–236.
Rendu, Q. , Philit, M. , Rozenberg, Y. , Aubert, S. , and Ferrand, P. , 2015, “ Regularization of Wilcox (2006) kω Turbulence Model and Validation on Shock-Wave/Boundary Layer Interaction,” 12th International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows (ISAIF), Genova, Italy, July 13–16, Paper No. 119. https://www.researchgate.net/publication/283576585_Regularization_of_Wilcox_2006_k-o_turbulence_model_and_validation_on_shock_waveboundary_layer_interaction
Corral, R. , and Vega, A. , 2016, “ The Low Reduced Frequency Limit of Vibrating Airfoils—Part I: Theoretical Analysis,” ASME J. Turbomach., 138(2), p. 021004. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Sketch of the flow in transonic KTH-VM100 facility

Grahic Jump Location
Fig. 2

Sketch of the polyurethane bump with camshaft

Grahic Jump Location
Fig. 3

Mesh deformation due to bump displacement: rigid body motion is imposed close to the bump (small cells) and mesh deformation is achieved in freestream part of the flow where cells are larger, involving small relative volume variation (<1%)

Grahic Jump Location
Fig. 4

Amplitude of bump deformation along streamwise axis, experimental data from Andrinopoulos et al. [14]

Grahic Jump Location
Fig. 5

Steady and extremes positions of the bump in URANS computations (corresponding to a maximum section variation of 0.5%)

Grahic Jump Location
Fig. 6

Structured H mesh of KTH-VM100 bi-dimensional bump

Grahic Jump Location
Fig. 7

Contours of Mach number over the transonic bump and position of separation (S) and reattachment (R) points obtained from steady computation

Grahic Jump Location
Fig. 8

Steady pressure coefficient along the bump: numerical results compared with experimental results of Ferria [12] and Bron [27]

Grahic Jump Location
Fig. 9

Amplitude and phase of the first harmonic of unsteady pressure coefficient along the bump at kr = 0.05 (f = 100 Hz) for nonlinear URANS and LRANS computations as well as experimental results of Ferria [12]

Grahic Jump Location
Fig. 10

Local aerodynamic work along the bump at kr = 0.05 (f = 100 Hz) for LRANS and nonlinear URANS computations

Grahic Jump Location
Fig. 11

Damping parameter along reduced frequency for the bump motion, obtained from LRANS computations

Grahic Jump Location
Fig. 12

Local aerodynamic work along the bump at kr = 0.15 (f = 300 Hz) for LRANS computation and nonlinear URANS computations

Grahic Jump Location
Fig. 13

Amplitude (top) and phase (bottom) of the first harmonic of unsteady pressure coefficient along the bump for linearized and nonlinear computations at kr = 0.15 (f = 300 Hz) for an amplitude of back pressure perturbations δPs/Psoutlet = 2% and a phase shift ΔΦ = 0 deg

Grahic Jump Location
Fig. 14

Amplitude (top) and phase (bottom) of the first harmonic of unsteady pressure coefficient along the bump for linearized and nonlinear computations at kr = 0.15 (f = 300 Hz) for an amplitude of back pressure perturbations δPs/Psoutlet = 2% and a phase shift ΔΦ = 90 deg

Grahic Jump Location
Fig. 15

Amplitude of unsteady static pressure due to bump motion (a) and acoustic waves (b), kr = 0.15 (f = 300 Hz)

Grahic Jump Location
Fig. 16

Phase of unsteady static pressure due to bump motion (a) and acoustic waves (b), kr = 0.15 (f = 300 Hz)

Grahic Jump Location
Fig. 17

Local aerodynamic work extracted along the bump at kr = 0.15 (f = 300 Hz) due to bump motion and back-pressure fluctuations at different phase shifts ΔΦ

Grahic Jump Location
Fig. 18

Influence of amplitude and phase shift of acoustic perturbations at kr = 0.15 (f = 300 Hz) on the added damping parameter (lines from linearized computations)

Grahic Jump Location
Fig. 19

Added damping parameter and phase measured at xC = 115 mm and xR = 201 mm along phase shift of back-pressure perturbations (kr = 0.15, f = 300 Hz)

Tables

Table Grahic Jump Location
Table 1 Parameters of bump deformation

Errata

Discussions

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