Adjoint Error Estimation and Adaptive Refinement for Embedded-Boundary Cartesian Meshes

The present dissertation deals with the application of an adaptive mesh hp-refinement method to the time-dependent compressible Euler equations describing the flow of an inviscid fluid. The physical simulation space is a two-dimensional channel with bump through which an ideal gas flows. A big advantage of the Euler equations is the applicability to different Mach regimes. Thus, smooth solutions and those containing discontinuities can be simulated. As the underlying numerical technique, a pseudo-spectral element method using least squares is used. The grid consists of rectangles that have to be transformed into curved deformed elements in the area of the bump. The h-refinement is realized by means of a quadtree and the p-refinement by means of a stepwise increase of the polynomial degree by 2, whereby hanging nodes can arise. Compared to previous research, the Euler equations are examined with respect to four different refinement criteria in combination with the hp-adaptive pseudo-spectral element method. The refinement criteria are the gradient criterion, two spectral criteria and an edge detection method. For a first determination of the suitability of the criteria for the h-refinement and for a validation of the pseudo-spectral element method, these are used for the solution of three steady state convection equations for which the exact solutions are known. Finally, the full method is applied to three test problems for the Euler equations. The flow through the channel is realized by a smooth subsonic flow, a transonic flow with a shock, and a supersonic flow that contains reflective shocks. Since no analytical solution is known, we compare our results to other works that consider stationary grids for these test cases. It turns out that the applied adaptive method refines the grid in the regions of the stagnation points and those that have discontinuities. Thus prior knowledge about the location of the problem areas is not necessary. A smaller number of elements and locally higher polynomial degrees reduce the sizes of the systems of equations to be solved.

A finite element method (FEM) for solution of the conservative full potential flow equation is presented as a means for improving the computation of transonic flows about complex geometries. The method does not require an orthogonal mesh, thus removing a major constraint in grid generation. A standard Galerkin finite element formulation is used in conjunction with the artificial compressibility method which stabilizes the algorithm in transonic flow and allows the capture of embedded shock waves. Solution comparisons, including one test case which stresses the'capabilities of state-of-the-art finite difference methods, verify the FEM accuracy and geometric capability. I. Introduction W HILE the computation of inviscid two- and threedimensional flowfields by finite difference methods has now become commonplace in industry, one of the most important technological deficiencies is in the area of geometric modeling, particularly for transonic flows. The finite element method (FEM) is presented as a way of alleviating some problems of geometric modeling which limit the effectiveness of current finite difference methods. Certain specific advantages of the FEM, if fully exploited, may improve greatly the computational state-of-the-art. Since the governing equations of inviscid transonic potential flow are nonlinear, the method of superposition (i.e., panel methods) is insufficient to yield a proper solution, even though solutions have been obtained by a method combining surface singularities and field differences.1 Also, because of the nonlinearity, similarity laws do not exist which allow subsonic flows to be related to transonic flows. Field methods must be used, typically finite difference or finite element methods, requiring a grid of points to be defined, both on the body surface arid in the flowfield, on which solutions to the governing equations are approximated. Finite difference method (FDM) computer codes generally

ABSTRACTThis article presents a new element-based finite-volume discretization approach for the solution of incompressible flow problems on co-located grids. The proposed method, called the method of proper closure equations (MPCE), employs a proper set of physically relevant equations to constrain the velocity and pressure at integration points. These equations provide a proper coupling between the nodal values of pressure and velocity. The final algebraic equations are not segregated in this study and are solved in a fully coupled manner. To show the applicability and performance of the method, it is tested on several steady two-dimensional laminar-flow benchmark cases. The results indicate that the method simulates the fluid flow in complex geometries and on nonorthogonal computational grids accurately. Also, it is shown that the method is robust in the sense that it does not require severe underrelaxation even at relatively high Reynolds numbers. In each test case, the required underrelaxation parameter, the number of iterations, and the corresponding CPU time are reported.

HemeLB: A high performance parallel lattice-Boltzmann code for large scale fluid flow in complex geometries

Large scale unsteady motions in many practical engineering flows play a very important role and it is very unlikely that these unsteady flow features can be captured within the framework of Reynolds averaged Navier–Stokes approach. Large‐eddy simulation (LES) has become, arguably, the only practical numerical tool for predicting those flows more accurately since it is still not realistic to apply DNS to practical engineering flows with the current and near future available computing power.Numerical methods for the LES of turbulent flows in complex geometry have been developed and applied to predict practical engineering flows successfully. The method is based on body‐fitted curvilinear co‐ordinates with the contravariant velocity components of the general Navier–Stokes equations discretized on a staggered orthogonal mesh. For incompressible flow simulations the main source of computational expense is due to the solution of a Poisson equation for pressure. This is especially true for flows in complex geometry. A multigrid 3D pressure solver is developed to speed up the solution. In addition, the Poisson equation for pressure takes a simpler form with no cross‐derivatives when orthogonal mesh is used and hence resulting in increased convergence rate and producing more accurate solutions. Copyright © 2004 John Wiley & Sons, Ltd.

An immersed boundary method for direct and large eddy simulation of stratified flows in complex geometry

A strongly conservative finite-volume procedure is presented for flows in complex geometries. The technique is based on a complete transformation of the governing equations, and physical velocity components, rather than the traditionally used Cartesian velocity components, are used as primitive variables. It was found that projecting the discretized vector transport equation in the direction of the covariant base vectors eliminated two substantial difficulties associated with flows in complex geometries. These difficulties stem from the presence of cross-pressure gradient terms and the need for a transformation between the different types of curvilinear velocity components in the mass conservation equation. It is shown that the present formulation ensures that the computational scheme is diagonally dominant. It was found that partially implicit treatment of nonorthogonal diffusion terms improved the convergence rate primarily for high-cell-Reynolds-number values. For nonstaggered grids, a new sol...

The purpose of this work is to introduce and validate a new staggered control volume method for the simulation of 2D/axisymmetric incompressible flows. The present study introduces a numerical procedure for solving the Navier–Stokes equations using the primitive variable formulation. The proposed method is an extension of the staggered grid methodology to unstructured triangular meshes for a control volume approach which features ease of handling of irregularly shaped domains. Two alternative elements are studied: transported scalars are stored either at the sides of an element or at its vertices, while the pressure is always stored at the centre of an element. Two interpolation functions were investigated for the integration of the momentum equations: a skewed mass-weighted upwind function and a flow-oriented exponential shape function. The momentum equations are solved over the covolume of a side or of a vertex and the pressure–velocity coupling makes use of a localized linear reconstruction of the discontinuous pressure field surrounding an element in order to obtain the pressure gradient terms. The pressure equation is obtained through a discretization of the continuity equation which uses the triangular element itself as the control volume. The method is applied to the simulation of the following test cases: backward-facing step flow, flow over a two-dimensional obstacle and flow in a pipe with sudden contraction of cross-sectional area. All numerical investigations are compared with experimental data from the literature. A grid convergence and error analysis study is also carried out for flow in a driven cavity. Results compared favourably with experimental data and so the new control volume scheme is deemed well suited for the prediction of incompressible flows in complex geometries. © 1997 John Wiley & Sons, Ltd.

A generalized finite-volume framework for the solution of fluid flows at all speeds in complex geometries and on unstructured meshes is presented. Starting from an existing pressure-based and fully-coupled formulation for the solution of incompressible flow equations, the additional implementation of pressure–density–energy coupling as well as shock-capturing leads to a novel solver framework which is capable of handling flows at all speeds, including quasi-incompressible, subsonic, transonic and supersonic flows. The proposed numerical framework features an implicit coupling of pressure and velocity, which improves the numerical stability in the presence of complex sources and/or equations of state, as well as an energy equation discretized in conservative form that ensures an accurate prediction of temperature and Mach number across strong shocks. The framework is verified and validated by a large number of test cases, demonstrating the accurate and robust prediction of steady-state and transient flows in the quasi-incompressible as well as subsonic, transonic and supersonic speed regimes on structured and unstructured meshes as well as in complex domains.

Wind-tunnel experiment plays a critical role in the design and development phases of modern aircraft, which is always limited by prohibitive cost. In contrast, numerical simulation, as an important alternative research paradigm, mimics complex flow behaviors but is less accurate compared to experiments. This leads to the recent development and emerging interest in applying data fusion for aerodynamic prediction. In particular, the accurate prediction of aerodynamics with lower computational cost can be achieved by fusing experimental (high-fidelity) and computational (low-fidelity) aerodynamic data. Currently, existing works on aerodynamic models using data fusion mainly concern integrated data, like lift, drag, etc. In this paper, a multifidelity aerodynamic model based on deep neural network (DNN) is developed for aerodynamic distribution over the wing surface, where both numerical and experimental data are introduced in the loss function with a proper weighting factor to balance the overall accuracy. The proposed approach is illustrated by modeling the surface pressure distribution of the ONERA M6 wing in transonic flow, including different scenarios where the flow condition varies or a small number of high-fidelity data are available. The results demonstrate that the proposed model trained based on a decent number of low-fidelity data and a few high-fidelity data can accurately predict the surface pressure distribution on the transonic wing. The outperformance of the proposed model over other DNNs with a single fidelity has been reported.

This paper builds on the computational aeroelastic results published previously and generated in support of the second Aeroelastic Prediction Workshop for the NASA Benchmark Supercritical Wing (BSCW) configuration. The computational results are obtained using FUN3D, an unstructured grid Reynolds-Averaged Navier-Stokes solver developed at the NASA Langley Research Center. The analysis results show the effects of the temporal and spatial resolution, the coupling scheme between the flow and the structural solvers, and the initial excitation conditions on the numerical flutter onset. Depending on the free stream condition and the angle of attack, the above parameters do affect the flutter onset. Two conditions are analyzed: Mach 0.74 with angle of attack 0 and Mach 0.85 with angle of attack 5. The results are presented in the form of the damping values computed from the wing pitch angle response as a function of the dynamic pressure or in the form of dynamic pressure as a function of the Mach number.

Sensitivity Derivatives for Advanced CFD Algorithm and Viscous Modeling Parameters via Automatic Differentiation

A computational method for the treatment of solid fluid interaction (SOFIA) has been developed to study the aeroelastic characteristics of wings and rotors by direct numerical simulation. It consists of an aerodynamic part to compute the airloads and a structural part to determine the structural deformation. The unsteady, three-dimensional flow is modeled by the Euler equations and calculated using an implicit solver. The elasticity is modeled by a generalized Timoshenko-Flugge beam, which includes the static and dynamic coupling of bending and torsional modes. A finite-element-method is implemented to predict the deformation. Both solvers are coupled in the time domain. SOFIA has been used to investigate a realistic Fokker-type wing in transonic flow. The steady lift distributions have been calculated for the rigid and the elastic wing. Comparison shows remarkable differences between wings with sweep-back and wings with no sweep.

Various engineering-type methods to calculate unsteady airloads on wings in transonic flow were applied in flutter calculations for a semispan flutter model of a supercritical wing. Verification was performed on the basis of comparing flutter characteristics, in which special attention was given to the prediction of transonic dips in the flutter boundaries. The methods were able to produce useful results when applied complementarity. ^0.95 Nomenclature b = dimensionless shift of aerodynamic center c =wing section chord, m C/n = sectional mean lift coefficient pressure coefficient at 0.95 chord, 0.7 span, upper side / = frequency, Hz k = reduced frequency related to semichord ka,ma = sectional unsteady lift and moment coefficients (AGARD notation, m related to 0.25 chord); kij.my = sectional unsteady lift and moment influence coefficients (induced by strip j, at strip /) MOO = Mach number P0 —

Numerical simulation of a square cavity was conducted to validate an implementation of an Immersed Boundary Method (IBM). The cavity consists of two differentially heated side walls and adiabatic top and bottom walls. A Cartesian grid is used with a finite volume, fractional step pressure correction method. Simulations use Dirichlet boundaries for vertical walls and Neumann boundaries for horizontal walls. The Immersed Boundary Method involves modifying the Navier--Stokes equations to include a forcing function in the momentum and energy equations that creates a virtual boundary. This method is useful because the boundary does not necessarily have to coincide with grid points; however, it is much less computationally expensive than other similar methods such as the cut cell method. The IBM is commonly used in simulations involving complex objects and can also accommodate moving boundaries. A standard numerical simulation with grid aligned with the boundary is first compared with previous results. The same geometry is then simulated by tilting the grid at various angles and using the IBM for each of the walls, and comparing these results with those initially obtained. We detail the implementation method and common problems associated with this. Velocity and temperature profiles are presented and the IBM is shown to maintain second order spatial accuracy. References E. A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. Journal of Computational Physics, 161(1):35--60, 2000. doi:10.1006/jcph.2000.6484 T. Gao, Y. H. Tseng, and X. Y. Lu. An improved hybrid cartesian/immersed boundary method for fluid-solid flows. International Journal for Numerical Methods in Fluids, 55(12):1189--1211, 2007. doi:10.1002/fld.1522 M. P. Kirkpatrick and S. W. Armfield. Experimental and large eddy simulation results for the purging of salt water from a cavity by an overflow of fresh water. International Journal of Heat and Mass Transfer, 48(2):341--359, 2005. doi:10.1016/j.ijheatmasstransfer.2004.08.016 M. P. Kirkpatrick, S. W. Armfield, and J. H. Kent. A representation of curved boundaries for the solution of the navier-stokes equations on a staggered three-dimensional cartesian grid. Journal of Computational Physics, 184(1):1--36, 2003. doi:10.1016/S0021-9991(02)00013-X B. P. Leonard and Simin Mokhtari. Beyond first-order upwinding: The ultra-sharp alternative for non-oscillatory steady-state simulation of convection. International Journal for Numerical Methods in Engineering, 30(4):729--766, 1990. doi:10.1002/nme.1620300412 S. H. Peng and L. Davidson. Large eddy simulation for turbulent buoyant flow in a confined cavity. volume 22, pages 323--331, 2001. doi:10.1016/S0142-727X(01)00095-9 C. S. Peskin. Numerical-analysis of blood-flow in heart. Journal of Computational Physics, 25(3):220--252, 1977. doi:10.1016/0021-9991(77)90100-0 J. Salat, S. Xin, P. Joubert, A. Sergent, F. Penot, and P. Le Quere. Experimental and numerical investigation of turbulent natural convection in a large air-filled cavity. International Journal of Heat and Fluid Flow, 25:824--832, 2004. doi:10.1016/j.ijheatfluidflow.2004.04.003 Y. H. Tseng and J. H. Ferziger. A ghost-cell immersed boundary method for flow in complex geometry. Journal of Computational Physics, 192(2):593--623, 2003. doi:10.1016/j/jcp.2003.07.024 N. Zhang and Z. C. Zheng. An improved direct-forcing immersed-boundary method for finite difference applications. Journal of Computational Physics, 221(1):250--268, 2007. doi:10.1016/j.jcp.2006.06.012 N. Zhang, Z. C. Zheng, and S. Eckels. Study of heat-transfer on the surface of a circular cylinder in flow using an immersed-boundary method. International Journal of Heat and Fluid Flow, 29(6):1558--1566, 2008. doi:10.1016/j.ijheatfluidflow.2008.08.009