Computation of Fluid Structure Interaction with Application to Three-Dimensional Combustion System Zinedine Khatir School of Mechanical and Aerospace Engineering Queen’s University Belfast Ashby Building, Stanmillis Road Belfast BT9 5AG, Northern Ireland, UK.
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[email protected] Summary. The computation of many fluid-structure interaction (FSI) problems requires solving simultaneously the coupled fluid and structure equations with an appropriate set of interface boundary conditions. When loosely-coupled partitioned procedures are considered in realistic situations where the fluid and structure subproblems have non-matching meshes, it is necessary to transfer the fluid pressure and stress fields at the fluid-structure interface into a structural load, and to convert the structural motion to the fluid system. For that purpose, an interpolation methodology based on volume spline is applied for transferring data between the fluid and structure grids. Results are presented with application to combustion system. Key words: Fluid Structure Interaction, CFD-CSD Coupling, 3-Dimensional Modelling, Non-Matching Meshes, Combustion System.
1 Introduction The numerical simulation of FSI problems occurs in many engineering and scientific applications, ranging from airfoil oscillations [1] or aero-hydrodynamics, haemodynamcis [2] to blade flutter analysis in turbomachinery [3] and power generation when designing gas turbine combustors [4, 5] for instance. Unlike tightly-coupled approaches where the fluid and structural equations are solved in a single computational domain, loosely-coupled partitioned procedures [6, 7] use independent techniques for the fluid and structure subdomains and exchange data along the fluid-structure interface. The loosely-coupled approach can thus take the full advantage of existing, well developed and tested codes for fluid and structural analysis. In this article, we are interested in enforcing the interface boundary condition by transferring the fluid loads to the structure, while displacement is transferred back to the fluid from the structure [8]. Various methods have been developed for the problem of data transfer with non-matching meshes [9]. A detailed review of data transfer methods is given in [10], which can be classified as methods suitable for the transfer of: (i) load only, (ii) displacement only and (iii) both load and displacement.
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The volume spline interpolation method by Hounjet et al [11] used for the data transfer in the present paper belongs to the third class. After a brief description of the method, the accuracy of the interpolation is assessed against simple 2D and 3D test cases. We will then apply the methodology for the application of FSI in threedimensional combustion system, and the numerical results show good agreement with experimental data.
2 Fluid-Structure Coupling 2.1 Principle The evaluation of the structure load induced by the fluid and the displacement of the CFD grid induced my the structural motion needs to satisfy the requirement of conservation of work and accuracy. The principle of virtual work is usually used to ensure conservation of energy. For this purpose, a linear transformation is sought. If the displacement ∆xa of the CFD grid is expressed by means of the structural displacement ∆xs using the transformation matrix [G] such that: ∆xf = [G] ∆xs
(1)
then the requirement for conservation leads to a corresponding matrix for the transformation of forces: fsT ∆xs = ffT ∆xf = ffT [G] ∆xs (2) fs = [G]T ff .
(3)
where ff and fs denote the fluid and structural loads respectively.
2.2 Volume Spline Interpolation Volume spline interpolation is a very simple method to interpolate non-uniformly spaced data. The method belongs to the category of scattered interpolation methods using radial basis functions [12, 13] and is well suited for transferring both load and displacement. Consider a set of unordered support points xi , i = 1 . . . N with functions values {fi ∈ R}, = 1 . . . N , and the interpolation function defined as: f (x) = α0 +
N X
αi Ri (x)
(4)
i=1
with Ri (x) =k x − xi k. The coefficients αi , i = 0 . .P . N are computed from the conditions f (xi ) = fi together with the side condition N i=1 αi = 0. Let us recall that the Laplacian of a scalar function φ(R) with R =k x k obeys ∂φ 1 ∂ 2 in 3D the relation ∆φ = R (R2 ∂R ). For φ(R) = R we have ∆φ = R and thus ∂R2 ∆∆φ = 0. Accordingly, the volume spline interpolation function f (x) verifies the biharmonic equation everywhere except at the support points. Therefore, the interpolation function f (x) will not possess new local extrema outside the support points [14]. This important property makes the volume spline method suitable, as it will reduce the risk of the occurrence of grid folding in the deformed mesh. The transformation matrix [G] as defined in Eq. (1) is computed as follows. Let us first define [G]∗ = [A][C]−1 where matrices [A] and [C] are constructed as:
3D Fluid Structure Interaction Computation 2
0 1 1 s s 6 1 E11 E 12 6 s s 6 [C] = 6 1 E21 E22 6. . . .. 4 .. .. s s 1 EN s 1 E N s 2
3 2 ... 1 a a 1 E11 E12 s . . . E1N s 7 a a 7 6 s 6 1 E21 E22 . . . E2N s 7 7 and [A] = 6 . . . .. .. 7 4 .. .. .. 5 . . a a 1 EN a 1 EN a 2 s . . . EN sNs
3 a . . . E1N s a . . . E2N s 7 7 7 .. .. 5 . . a . . . EN a N s
3
(5)
where N a and N s are the number of fluid and structural elements respectively. The functions E a and E s are given by p s (xl − xm )2 + (yl − ym )2 + (zl − zm )2 Elm = (6) p a Elm = (xal − xm )2 + (yla − ym )2 + (zla − zm )2 with (x, y, z) and (xa , y a , z a ) are the coordinates of the structural points and fluid points respectively. The coupling matrix [G] is then obtained by deleting the first column of [G]∗ .
3 Numerical Results Let us first consider a 2D and 3D simple tests followed by a combustion application.
3.1 2D Simple Test The interface between CFD and CSD consists of the unit circle centered at the origin. As seen in Fig. 1-a the two meshes are non-matching. The aerodynamic input field points is obtained from 30 discretized points on the unit circle, whereas the structural input field is taken from 23 discretized points on the circle with radius 1.05. After that, the following deformation is applied to the aerodynamic input points (xi , yi ) to obtain the new position (x∗i , yi∗ ): •
For i=1 to 30 × (i − 1) θ = 2π 30 x∗i = r cos θ = xi , yi∗ = r sin θ = yi If xi > 0.8: x∗i = xi − 0.8 ∗ x3i End
The final solution is depicted in Fig. 1-b after interpolation. The deformation of the structural mesh is of very good quality. The interpolated solution is compared with the exact solution. The maximum obtained error is around 0.07 which gives second order precision for this test E ' (δx)2 .
Fig. 1. Two circular concentric non-matching meshes, (dots) CFD mesh and (square) CSD mesh configuration: (a) Before deformation and (b) After deformation.
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3.2 3D Simple Test Here a random pressure field is applied onto the unstructured CFD mesh as in Fig. 2. This pressure field is then interpolated onto the structured CSD mesh. As shown in Fig. 2, the input random pressure field and the interpolated pressure field on the CSD mesh compare well. The maximum obtained error is E ' 0.008 which leads to a third order precision with this kind of discretization.
(a) (b) Fig. 2. (a) CFD and (b) CSD non-matching meshes alongside pressure fluctuation.
3.3 Combustion System Problem Many combustors exhibit combustion instabilities which constitute significant risks for project developments [15]. The prediction of combustion stability in a specified burner is the focus of either experimental [16] or numerical investigations [17]. Of crucial importance for the operation of the engine combustors is its structural integrity. This may cause the liner to vibrate excessively and could be dangerous. It is thus important to understand transient combustion an its coupling with wall vibration in typical combustion chamber. A typical experimental setup together with computational domain used for the numerical simulation is shown if Fig.3
Fig. 3. (Left) Experimental setup and (Right) Computational domain [5, 17]. A transient FSI calculation is implemented to assessed volume spline method for data transfer. The LES solver AVBP (i.e. www.cerfacs.fr/cfd/CFDWeb.html) simulates the full compressible multi-species Navier-Stokes equations and provides the pressure loads needed for the FEM structural code solver CalculiX (i.e. www.calculix.de). The interface non-matching meshes is shown in Fig.4. The LES
Fig. 4. Fluid and structure meshes with non-matching interfaces.
3D Fluid Structure Interaction Computation
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data are transferred onto the CSD grid every 0.15 ms with a total run of 21 ms. Fig. 5 display the computed and measured velocity on the liner. Results compare well. The time evolution of the loads is plotted in Fig. 6. CFD and CSD loads after transfer coincide. Fig. 7 demonstrate the energy conservation during the simulation.
Fig. 5. Experiment,
, and calculated ,− − −, velocity on the liner [5, 17].
Fig. 6. Time evolution of the sum of the CFD, on the vibrating liner.
& CSD, −−− loads respectively
Fig. 7. Time evolution of 1st (Top) and 2nd (Bottom) Moment (w.r.t axis x=0.895 m) and relative errors. A mean error of 1.26×10−5 and 9.01×10−4 is found for the time evolution of the 1st and 2nd moment respectively. This proves again an accurate data transfer between the 2 non-matching grids as plotted in Fig. 8.
Fig. 8. Surface plot of CFD (Left) and CSD (Right) loads at time t=0.485s.
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4 Conclusion A volume spline method for the data transfer in transient FSI computation with nonmatching meshes has been presented. Results show good agreement with analytical and experimental results. Further investigations are still being undertaken to provide a better understanding of FSI in combustion systems. Acknowledgement. This work is carried out within the EC Framework Marie Curie RTN project FLUISTCOM-Fluid-Structure Interaction for Combustion Systems through contract number MRTN-CT-2003-504183. The author wishes to thank A.X. Sengissen and P. Mauro from CERFACS - France, R. Huls and A.K. Pozarlik from Univ. of Twente - Netherlands their constant help and assistance. He also expresses his gratitude to Dr R. Cooper and Dr J. Watterson for their help and comments on the paper. The invaluable help of N. Forsythe at Queen’s Univ. Belfast is greatly appreciated.
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