Interactive computing methods for aeroelastic analysis of turbomachinery
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Interactive computing methods for aeroelastic analysis of turbomachinery Alexandru Dumitrache * Institute of Statistics and Applied Mathematics, P.O. Box 1-24, RO-010145, Bucharest, Romania An interaction viscous-inviscid method for efficiently computing steady and unsteady viscous flows is presented. The inviscid domain is modeled using a finite element discretization of the full potential equation. The viscous region is modeled using a finite difference boundary layer technique. The two regions are simultaneously coupled using the transpiration approach. A time linearization technique is applied to this interactive method. For unsteady flows, the fluid is assumed to be composed of a mean or steady flow plus a harmonically varying small unsteady disturbance. Numerically exact nonreflecting boundary conditions are used for the far field conditions. Results for some steady and unsteady, laminar and turbulent flow problems are compared to linearized Navier-Stokes or time-marching boundary layer methods. 1 Introduction A natural solution for computing the unsteady flow through cascades, related to aeroelastic phenomena, would be to use a time- marching Navier-Stokes CFD solution. This approach is prohibitively expensive, particularly for routine design use. As an alternative, one can use in our computing an interactive viscous-inviscid (IVI) method, completed with a fully simultaneously coupling mechanism. The outer (inviscid) flow field is modeled using a finite element discretisation of the full potential (FP) equation. The non-linear time dependent boundary-layer (BL) equations (inner region) are linearized about the steady solution to obtain the linearized unsteady equations. The unsteady perturbations are assumed to be harmonic in time. 2 Flow field description Inviscid Flow. Firstly we will consider the steady and unsteady flow in a channel, using IVI method. The inviscid (outer) flow is assumed to be isentropic, irrotational, and two-dimensional. Hence, the outer field may be represented by a scalar veloc- ity potential, ˆ φ. Using the equation of conservation of mass (compressible potential flow ) and substituting the density ˆ ρ and the pressure ˆ p in terms of the velocity potential, one obtains the FP equation, ∇ 2 ˆ Φ= 1 ˆ c 2 • ∂ 2 ˆ Φ ∂t 2 +2∇ ˆ Φ ·∇ ∂ ˆ Φ ∂t + 1 2 ∇ ˆ Φ ·∇ ‡ ∇ ˆ Φ · 2 ‚ , where c is the local speed of sound. This differential equation is the Euler-Lagrange equation of a variational principle (Bate- man [1]; Hall [2]), with the natural BC, ˆ ρ ∂ ˆ φ ∂n = ˆ Q, where ˆ Q is the mass flux. Viscous Flow. The viscous region next to the solid surface or in the wake of airfoil is modeled by incompressible (BL) equation. For incompressible flow, the continuity equations derived from the mass conservation reduces to ∇· ‡ ∇ ˆ φ · =0. Starting from the Navier-Stokes equations, after some calculations and eliminating the pressure ˆ p, one obtain the s-momentum equation at the edge of the BL: ∂ ˆ u ∂t +ˆ u ∂ ˆ u ∂s +ˆ v ∂ ˆ u ∂n = ∂ ˆ u e ∂t +ˆ u e ∂ ˆ u e ∂s + μ ˆ ρ ∂ 2 ˆ u ∂n 2 , (2.1) with three BC: ˆ u =ˆ v =0 at n =0, (no-slip condition) and ˆ u =ˆ u e as n →∞. Here s and n are coordinates parallel and normal to the boundary, and ˆ u and ˆ v are the velocity components in the parallel and normal directions. Coordinate Transformation. The approach used here is to linearize the FP (nonlinear, unsteady) equation about some nominal mean flow. One can suppose that the unsteadiness of the flow in cascaded airfoils is small compared to the mean flow and that the unsteady part of the flow is harmonic in time, which removes the explicit time dependency from the FP equation. Instead of solving the BL equations written with the ˆ u, ˆ v variables, one follows Blasius modified coordinate transformation and introduces a stream function ˆ Ψ, which varies not only in the n-direction but also in time and in the n-direction, ˆ Ψ(s, n, t)= √ νsu 0 ˆ f (s, η, t) , where f is the vector of grid motion functions, (f, g) T . To reduce the BL grid variation along the airfoil and wake-cut line, a second coordinate transformation is introduced based on the mean value of the displacement thickness, ¯ δ. By defining ¯ η (s, n)= n/ ¯ δ (s) and ˆ Ψ(s, n)= f (s, η) ¯ δ, and after substituting these terms in the momentum equation (2.1), one obtains : ‡ ˆ b ˆ f 00 ) 0 + ¯ δ ¯ δ s ˆ f ˆ f 00 + ¯ δ 2 ( ∂ ˆ u e ∂t +ˆ u e ∂ ˆ u e ∂s ) = ¯ δ 2 ‡ ∂ ˆ f 0 ∂t + ˆ f 0 ∂ ˆ f 0 ∂s - ˆ f 00 ∂ ˆ f ∂s · . The unsteady BL equations are solved using the linearization approach and assuming harmonically varying small dis- turbances. After expanding the nondimensional stream function ˆ f and the edge velocity ˆ u e in the perturbation series: * Corresponding author: e-mail: [email protected], Phone: +40 21 411 4900, Fax: +40 21 411 4305 PAMM · Proc. Appl. Math. Mech. 4, 560–561 (2004) / DOI 10.1002/pamm.200410261 © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Transcript of Interactive computing methods for aeroelastic analysis of turbomachinery
Interactive computing methods for aeroelastic analysis of turbomachinery
Alexandru Dumitrache∗
Institute of Statistics and Applied Mathematics, P.O. Box 1-24, RO-010145, Bucharest, Romania
An interaction viscous-inviscid method for efficiently computing steady and unsteady viscous flows is presented. The invisciddomain is modeled using a finite element discretization of the full potential equation. The viscous region is modeled using afinite difference boundary layer technique. The two regions are simultaneously coupled using the transpiration approach. Atime linearization technique is applied to this interactive method. For unsteady flows, the fluid is assumed to be composedof a mean or steady flow plus a harmonically varying small unsteady disturbance. Numerically exact nonreflecting boundaryconditions are used for the far field conditions. Results for some steady and unsteady, laminar and turbulent flow problemsare compared to linearized Navier-Stokes or time-marching boundary layer methods.
1 Introduction
A natural solution for computing the unsteady flow through cascades, related to aeroelastic phenomena, would be to use a time-marching Navier-Stokes CFD solution. This approach is prohibitively expensive, particularly for routine design use. As analternative, one can use in our computing an interactive viscous-inviscid (IVI) method, completed with a fully simultaneouslycoupling mechanism. The outer (inviscid) flow field is modeled using a finite element discretisation of the full potential (FP)equation. The non-linear time dependent boundary-layer (BL) equations (inner region) are linearized about the steady solutionto obtain the linearized unsteady equations. The unsteady perturbations are assumed to be harmonic in time.
2 Flow field description
Inviscid Flow. Firstly we will consider the steady and unsteady flow in a channel, using IVI method. The inviscid (outer)flow is assumed to be isentropic, irrotational, and two-dimensional. Hence, the outer field may be represented by a scalar veloc-ity potential,φ. Using the equation of conservation of mass (compressible potential flow ) and substituting the densityρ and the
pressurep in terms of the velocity potential, one obtains the FP equation,∇2Φ = 1c2
[∂2Φ∂t2 + 2∇Φ · ∇∂Φ
∂t + 12∇Φ · ∇
(∇Φ
)2]
,
wherec is the local speed of sound. This differential equation is the Euler-Lagrange equation of a variational principle (Bate-
man [1]; Hall [2]), with the natural BC,ρ∂φ∂n = Q, whereQ is the mass flux.
Viscous Flow. The viscous region next to the solid surface or in the wake of airfoil is modeled by incompressible (BL)
equation. For incompressible flow, the continuity equations derived from the mass conservation reduces to∇ ·(∇φ
)= 0.
Starting from the Navier-Stokes equations, after some calculations and eliminating the pressurep, one obtain thes-momentumequation at the edge of the BL:
∂u
∂t+ u
∂u
∂s+ v
∂u
∂n=
∂ue
∂t+ ue
∂ue
∂s+
µ
ρ
∂2u
∂n2, (2.1)
with three BC:u = v = 0 at n = 0, (no-slip condition) andu = ue asn → ∞. Heres andn are coordinates parallel andnormal to the boundary, andu andv are the velocity components in the parallel and normal directions.
Coordinate Transformation. The approach used here is to linearize the FP (nonlinear, unsteady) equation about somenominal mean flow. One can suppose that the unsteadiness of the flow in cascaded airfoils is small compared to the mean flowand that the unsteady part of the flow is harmonic in time, which removes the explicit time dependency from the FP equation.Instead of solving the BL equations written with theu, v variables, one follows Blasius modified coordinate transformation andintroduces a stream functionΨ, which varies not only in then-direction but also in time and in then-direction,Ψ (s, n, t) =√
νsu0f (s, η, t) , wheref is the vector of grid motion functions,(f, g)T . To reduce the BL grid variation along the airfoiland wake-cut line, a second coordinate transformation is introduced based on the mean value of the displacement thickness,δ.
By definingη (s, n) = n/δ (s) andΨ (s, n) = f (s, η) δ, and after substituting these terms in the momentum equation (2.1),
one obtains :(bf ′′)′ + δ δs f f ′′ + δ2
(∂ue
∂t + ue∂ue
∂s
)= δ2
(∂f ′
∂t + f ′ ∂f ′
∂s − f ′′ ∂f∂s
).
The unsteady BL equations are solved using the linearization approach and assuming harmonically varying small dis-turbances. After expanding the nondimensional stream functionf and the edge velocityue in the perturbation series:
∗ Corresponding author: e-mail:[email protected], Phone: +40 21 411 4900, Fax: +40 21 411 4305
PAMM · Proc. Appl. Math. Mech. 4, 560–561 (2004) / DOI 10.1002/pamm.200410261
© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
f (s, η, t) = F (s, η) + f (s, η) ejωt, ue (s, t) = Ue (s) + ue (s) ejωt, and substituting into momentum equation, oneobtains the mean flow equation of the BL , collecting the zero-th-order terms, and the small disturbance BL equations, col-lecting the first order terms.
Boundary Conditions (BC). For inviscid flow the near-field BC include airfoil surface, wake and periodic BC. For steadyflow, the velocity normal to the airfoil surface is zero. For harmonically varying small disturbance unsteady flow, the airfoiltangency BC is a natural BC of the variational principle. For wake BC, steady or harmonically varying small disturbanceunsteady flow, the periodicity condition is imposed by requiring the continuity pressure. The incoming BC is usually onechord away from the airfoil leading edge and similarly, the downstream outgoing boundary of the computational domainis about one chord away from the trailing edge. For viscous flow one needs three BC for airfoil, written as functions ofnondimensional potential functionf and the new coordinate system variable(s, η). The initial conditions is a similaritysolution obtained by solving Blasius’ equation. Along the wake-cut, the two shear layers must be continuous, that is, theirvelocities and slopes must be continuous. For the laminar-turbulent transition theen method is chosen, and the algebraic eddyviscosity model of Cebeci and Smith was used.
2.1 Simultaneous coupling of the viscous and inviscid regions
The coupling has to assure the continuity of flow variable, such as velocity and pressure. In the fully simultaneous coupling(Veldman, [3]) of the differential BL, the value of the displacement thicknessδ∗ and edge velocityue are unknown. Theviscous shear layer is represented by a surface transpiration velocity at the airfoil surface and the wake-cut. No iterations arerequired, once the simultaneous set of equations is solved. The equivalent inviscid flow (EIF) is solved with the following BC:
Q = ρe
(∂δ∗∂t + ue
∂δ∗∂s
), where the mass fluxQ = ρe
∂φ∂n at n = 0 and ue = ∂φ
∂n
∣∣∣n=0
. To simulate presence of the viscous
region, the injection velocity,(= Dδ∗Dt ), is imposed at the edge of the EIF. For the harmonically varying small disturbance
unsteady flow, the modified inviscid flow BC which must be imposed along the airfoil and along the wake. A rectangular gridwas generated for the steady and for the unsteady boundary layer equations. By using the Newton iteration procedure, thenonlinear equations are transformed into a series of linear equations which are solved using the lower-upper decomposition.
Fig. 1 Unsteady boundary layer quantities on lower wall of diffuser.
3 Validation example and conclusion
To validate the computation of the small disturbance boundary layer flow, the unsteady viscous flow over a semi-infinite flatplate is compared to the exact theory (I) and to validate the capability of the viscous-inviscid interaction method of computingseparated flows, the flow in a diffuser is presented (II). In Figure 1 one can be shown the unsteady boundary layers flowswith separation and reattachments due to a unit perturbation in the outer flow exit velocity with a frequency of0.1 (a reducedfrequency of0.6). Lower portion of figure shows perturbation velocity profiles.
This model has been validated and shown to provide similar results compared with previous computational and experimen-tal data presented in the literature. A typical unsteady flow calculation, including the computation of the steady flow, requiresless than of 2 min CPU on a workstation computer.
Further, the method is able to predict unsteady aeroelastic phenomena such as stall flutter.
References
[1] H. Bateman, Irrotational motion of a compressible fluid, Proceeding of the National Academy of Science16, 816–825 (1930).[2] K. C. Hall, Deforming grid variational principle for unsteady small disturbances flows in cascades, AIAA Journal31, 891–900 (1993).[3] A. E. P. Veldman, New quasi-simultaneous method to calculate interacting boundary layers, AIAA Journal19, 79–85 (1981).
© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Section 17 561
