Residual Error Estimation And Adaptive Algorithms For Fluid Flows
Abstract
The thesis deals with the development of a new residual error estimator and adaptive algorithms based on the error estimator for steady and unsteady fluid flows in a finite volume framework. The aposteriori residual error estimator referred to as Rparameter, is a measure of the local truncation error and is derived from the imbalance arising from the use of an exact operator on the numerical solution for conservation laws. A detailed and systematic study of the Rparameter on linear and nonlinear hyperbolic problems, involving continuous flows and discontinuities is performed. Simple theoretical analysis and extensive numerical experiments are performed to establish the fact that the Rparameter is a valid estimator at limiterfree continuous flow regions, but is rendered inconsistent at discontinuities and with limiting. The Rparameter is demonstrated to work equally well on different mesh topologies and detects the sources of error, making it an ideal choice to drive adaptive strategies. The theory of the error estimation is also extended for unsteady flows, both on static and moving meshes. The Rparameter can be computed with a low computational overhead and is easily incorporated into existing finite volume codes with minimal effort.
Adaptive refinement algorithms for steady flows are devised employing the residual error estimator. For continuous flows devoid of limiters, a purely Rparameter based adaptive algorithm is designed. A threshold length scale derived from the estimator determines the refinement/derefinement criterion, leading to a selfevolving adaptive algorithm devoid of heuristic parameters. On the other hand, for compressible flows involving discontinuities and limiting, a hybrid adaptive algorithm is proposed. In this hybrid algorithm, error indicators are used to flag regions for refinement, while regions of derefinement are detected using the Rparameter. Two variants of these algorithms, which differ in the computation of the threshold length scale are proposed. The disparate behaviour of the Rparameter for continuous and discontinuous flows is exploited to design a simple and effective discontinuity detector for compressible flows. For timedependent flow problems, a twostep methodology is proposed for adaptive grid refinement. In the first step, the ``best" mesh at any given time instant is determined. The second step involves predicting the evolution of flow phenomena over a period of time and refines regions into which the flow features would progress into. The latter step is implemented using a geometricbased ``Refinement Level Projection" strategy which guarantees that the flow features remain in adapted zones between successive adaptive cycles and hence uniform solution accuracy. Several numerical experiments involving inviscid and viscous flows on different grid topologies are performed to illustrate the success of the proposed adaptive algorithms.
Appendix 1
Candidate's response to the comments/queries of the examiners
The author would like to thank the reviewers for their appreciation of the work embodied in the thesis and for their comments. The clarifications to the comments and queries posed in the reviews are summarized below.
Referee 1
Q: The example of mesh refinement for RANS solution with shock was performed with isotropic mesh, while the author claims that it is appropriate with anisotropic mesh. If this is the case, why did he not demonstrate that ? As the author knows well, in the case of full 3D configuration, isotropic adaptation will lead to substantial grid points. The large mesh will hamper timely turnaround time of simulation. Therefore it would be a significant contribution to the aero community if this point is investigated at a later date.
Response: The author is of the view that for most practical situations, a pragmatic approach to mesh adaptation for RANS computations would merely involve generating a viscous padding of adequate fineness around the body and allowing for grid adaptation only in the outer potential region. Of course, this method would allow for grid adaptation in the outer layers of viscous padding only to the extent the smoothness criterion is satisfied while adapting the grids in the potential region. This completely obviates point addition to the wall (CAD surface) and there by avoids all complexities (like loss in automation) resulting from the interaction with the surface modeler while adding point on the wall. This method is expected to do well for attached flows and mildly separated flows. This method is expected to do well even for problems involving shock  boundary layer interaction, owing to the fact that the shock is normal to the wall surface (recall, a flow aligned grid is ideal to capture such shocks), as long as the interaction does not result in a massive separation. This approach has already been demonstrated in section 4.5.3 where in adaptive highlift computations have been performed.
Isotropic adaptation retains the goodness of the zero level grid and therefore the robustness of the solver does not suffer through successive levels of grid adaptation. This procedure may result in large number of volumes. On the other hand, the anisotropic refinement may result in significantly less number of volumes, but the mesh quality may have badly degenerated during successive levels of adaptation leading to difficulties in convergence. Therefore, the choice of either of these strategies is effectively dictated by requirements on grid quality and grid size. Also, it is generally understood that building tools for anisotropic adaptation are more complicated as compared to those required for isotropic adaptation, while anisotropic refinement may not require point addition on the wall. Considering these facts, in the view of the author, this issue is an open issue and his personal preference would be to use isotropic refinement or a hybrid strategy employing a combination of these methodologies, particularly considering aspects of solution quality.
Finally, in both the examples cited by the reviewer (sections 6.4.5 & 6.4.6) the objective was to demonstrate the efficacy of the new adaptive algorithm (using error indicators and the residual estimator), rather than evaluating the pros & cons of isotropic and anisotropic refinement strategies. In the sections cited above, the author has merely highlighted the advantages of the refinement strategies in specific context of the problem considered and these statements need not be considered as general.
Referee 2
Q: For convection problems, a good error estimator must be able to distinguish between locally generated error and convected error. The thesis says the residual error estimator is able to do this and some numerical evidence is presented, but can the candidate comment how the estimator is able to achieve this ?
Response: The ultimate aim of any AMR strategy is to reduce the global error. The residual error estimator proposed in this work measures the local truncation error. It has been shown in the context of a linear convective equation that the global error in a cell consists of two partsthe locally generated error in the cell (which is the Rparameter) and the local error transported from other cells in the domain. Either of these errors are dependent on the local error itself and any algorithm that reduces the local truncation error (sources of error) will reduce the global error in the domain. This conclusion is supported by the test case of isentropic flow past an airfoil (Chapter 3, C, Pg 79), where refinement based on the Rparameter leads to lower global error levels than a global error based refinement itself.
Q: While analysing the Rparameter in Section 3.3, the operator δ2 is missing.
Response: The analysis in Section 3.3 is based on Eq.(3.3) (Pg 58) which provides the local truncation error. As can be seen from Eq.(3.14), the LHS represents the discrete operator acting on the numerical solution (which is zero) and the first term on the RHS is the exact operator acting on the numerical solution (which is I[u]). Consequently the truncation terms T1 and T2 contribute to the truncation error R1 . However, from the viewpoint of computing the error estimate on a discretised domain, we need to replace the exact operator I by a higher order discrete operator δ2 . This gives the Rparameter, which has contributions from R1 as well as discretisation errors due to the higher order operator, R2 . When the latter is negligible compared to the former, the Rparameter is an estimate of the local truncation error. The truncation error depends on the accuracy of the reconstruction procedure used in obtaining the numerical solution and hence on the discrete operator δ1. On very similar lines, it can be shown that operator δ2 leads to a formal second order accuracy and this operator is only required in computing the residual error estimate.
Q: What does the phrase "exact derivatives of the numerical solution" mean ?
Response: This statement exemplifies the fact that the numerical solution is the exact solution to the modified partial differential equation and that the truncation terms T1 and T2 that constitute the Rparameter are functions of the derivatives of this numerical solution.
Q: For the operator δ2 quadratic reconstruction is employed. Is the exact or numerical flux function used ?
Response: The operator δ2 is a higher order discrete approximation to the exact operator I. Therefore, a quadratic polynomial with a threepoint Gauss quadrature has been used in the error estimation procedure. Error estimation does not involve issues with convergence associated with the flow solver and therefore an exact flux function has been employed with the δ2 operator. Nevertheless, it is also possible to use the same numerical flux function as employed in the flow solver for error estimation also.
Q: The same stencil of grid points is used for the solution update and the error estimation. Does this not lead to an increased stencil size ?
Response: In comparison to reconstruction using higher degree polynomials such as cubic and quartic reconstruction, quadratic reconstruction involves only a smaller stencil of points consisting of the nodesharing neighbours of a cell. The use of such a support stencil is sufficient for linear reconstruction also and adds to the robustness of the flow solver, although a linear reconstruction can, in principle, work with a smaller support stencil. A possible alternative to using quadratic reconstruction (and hence a slightly larger stencil) is to adopt a Defect Correction strategy to obtain derivatives to higher order accuracy and needs to be explored in detail.
Q: How is the Rparameter computed for viscous flows ?
Response: The computation of the Rparameter for viscous flows is on the same lines as for inviscid flows. The gradients needed for viscous flux computation at the face centers are obtained using quadratic reconstruction. The procedure for calculation of the Rparameter for steady flows (both inviscid and viscous) is the stepbystep algorithm in Section 3.5.
Q: In some cases, regions ahead of the shock show no coarsening.
Response: The adaptive algorithm proposed in this work does not allow for coarsening of the initial mesh, and regions ahead of the shock remain unaffected (because of uniform flow) at all levels of refinement.
Q: Do adaptation strategies terminate automatically atleast for steady flows ?
Response: The adaptation strategies (RAS and HAS) must, in principle by virtue of construction of the algorithm, automatically terminate for steady flows. In the HAS algorithms though, there are certain heuristic criteria for termination of refinement especially at shocks/turbulent boundary layers. In this work, a maximum of four cycles of refinement/derefinement have only been carried out and therefore an automatic termination of the adaptive strategies were no studied.
Q: How do residualbased adaptive strategies compare and contrast with adjointbased approaches which are now becoming popular for goaloriented adaptation ?
Adjointbased methods involve solution to the adjoint problem in addition to solving the primal problem, which represents a substantial computational cost. A timing study for a typical 3D problem[2] indicates that the solution of the adjoint problem (which needs the computation of the Jacobian and sensitivities of the functional) could require as much as onehalf of the total time needed to compute the flow solution. On the contrary, Rparameter based refinement involves no additional information than that required by the flow solver and is roughly equivalent to one explicit iteration of the flow solver (Section 3.5.1). For practical 3D applications, adjointbased approaches will lead to a prohibitively high cost, and more so for dynamic adaptation. This is also exemplified by the fact that there has been only few recent works on 3D adaptive computations based on adjoint error estimation (which consider only inviscid flows)[1,2].
Goaloriented adaptation involves reducing the error in some functional of interest. This can be achieved within the framework of Rparameter based adaptation, by introducing additional termination criteria based on integrated quantities. Within an automated adaptation loop, such an algorithm would terminate when the integrated quantities do not change appreciably with refinement levels. This is in contrast to the adjointbased approach which strives to reduce the error in the functional below a certain threshold. Considering the fact that reducing the residual leads to reducing the global error itself, the Rparameter based adaptive algorithm would also lead to accurate estimates of the integrated quantities (which depend on the numerical solution). This is also reflected in the fact that the Rparameter based adaptation for the threeelement NHLP configuration predicts the lift and drag coefficients to reasonable accuracy, as shown in Section 4.5.3.
The author is of the belief that the Rparameter based adaptive algorithm holds huge promise for adaptive simulations of flow past complex geometries, both in terms of computational cost and solution accuracy. This is exemplified by successful adaptive simulations of inviscid flow past ONERA M6 wing as well as a conventional missile configuration[3]. A more concrete comparison of the Rparameter based and adjointbased approaches would involve systematically solving a set of problems by both approaches and has not been considered in this thesis.
[1] Nemec and Aftosmis,``Adjoint error estimation and adaptive refinement for embeddedboundary cartesian meshes", AIAA Paper 20074187, 2007.
[2] Wintzer, Nemec and Aftosmis,``Adjointbased adaptive mesh refinement for sonic boom prediction", AIAA Paper 20086593, 2008.
[3] Nikhil Shende, ``A general purpose flow solver for Euler equations", Ph.D. Thesis, Dept. of Aerospace Engg., Indian Institute of Science, 2005.
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