Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations (original) (raw)
Related papers
Geometric conservation laws for aeroelastic computations using unstructured dynamic meshes
12th Computational Fluid Dynamics Conference, 1995
Numerical simulations of flow problems with moving boundaries commonly require the solution of the fluid equations on unstructured and deformable dynamic meshes. In this paper, we present a unified theory for deriving Geometric Conservation Laws (GCLs) for such problems. We consider several popular discretization methods for the spatial approximation of the flow equations including the Arbitrary Lagrangian-Eulerian (ALE) finite volume and finite element schemes, and space-time stabilized finite element formulations. We show that, except for the case of the space-time discretization method, the GCLs impose import ant constraints on the algorithms employed for time-integrating the semi-discrete equations governing the fluid and dynamic mesh motions. We address the impact of theses constraints on the solution of coupled aeroelastic problems, and highlight the importance of the GCLs with an illustration of their effect on the computation of the transient aeroelastic response of a flat panel in transonic flow.
Computers & Fluids
In this paper, we present a novel second-order accurate Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving nonconforming polygonal grids, in order to avoid the typical mesh distortion caused by shear flows in Lagrangian-type methods. In our new approach the nonconforming element interfaces are not defined by the user, but they are automatically detected by the algorithm if the tangential velocity difference across an element interface is sufficiently large. The grid nodes that are sufficiently far away from a shear wave are moved with a standard node solver, while at the interface we insert a new set of nodes that can slide along the interface in a nonconforming manner. In this way, the elements on both sides of the shear wave can move with a different velocity, without producing highly distorted elements.
This paper describes a finite volume solver for the computation of unsteady Euler flows on moving unstructured grids. The Arbitrary Lagrangian Eulerian formulation takes the grid velocity into account. A Cranck-Nicolson time integration scheme and a quadratric spatial reconstruction technique are applied and yield a second order time and space truncation errors. A new way of satisfying the Discrete Geometric Conservation Law adapted to this scheme is proposed. This new method is computational efficient as it leaves the number of fluxes evaluations equal to their fixed grid counterpart. Results highlighting the robustness and accuracy of the proposed approach are presented.
On the geometric conservation law in transient flow calculations on deforming domains
International Journal for Numerical Methods in Fluids, 2006
This note revisits the derivation of the ALE form of the incompressible Navier-Stokes equations in order to retain insight into the nature of geometric conservation. It is shown that the ow equations can be written such that time derivatives of integrals over moving domains are avoided prior to discretization. The geometric conservation law is introduced into the equations and the resulting formulation is discretized in time and space without loss of stability and accuracy compared to the ÿxed grid version. There is no need for temporal averaging remaining. The formulation applies equally to di erent time integration schemes within a ÿnite element context. Copyright ? 2005 John Wiley & Sons, Ltd.
Second-order implicit schemes that satisfy the GCL for flow computations on dynamic grids
36th AIAA Aerospace Sciences Meeting and Exhibit, 1998
We consider the solution of three-dimensional flow problems with moving boundaries using the Arbitrary Lagrangian Eulerian formulation or dynamic meshes. We focus on the case where spatial discretization is performed by unstructured finite volumes or finite elements. We formulate the consequence of the Geometric Conservation Law on the second-order implicit temporal discretization of the semi-discrete equations governing such problems, and use it as a guideline to construct a new family of second-order time-accurate and geometrically conservative implicit numerical schemes for flow computations on moving grids. We apply these new algorithms to the solution of three-dimensional flow problems with moving and deforming boundaries, demonstrate their superior accuracy and computational efficiency, and highlight their impact on the simulation of fluid/structure interaction problems.
We consider the conserv.ation properties of a staggered-grid Lagrange formulation of the hydrodynamics equations (SGH). Hydrodynamics algorithms are often formulated in a relatively ad hoc manner in which independent discretizations are proposedfor mass, momentum, energy, and so forth. We show that, once discretizations for mass and momentum are stated, the remaining discretizations are very nearly uniquely determined, so there is very liule latitude for variation. As has been known for some time, the kinetic energy discretization must follow directly from the momentum equation; and the internal energy must follow directly from the energy currents affecting the kinetic energy. A fundamental requirement (termed isentropicity)for numerical hydrodynamics algorithms is the ability to remain on an isentrope in the absence of heating or viscous forces and in the limit of small timesteps. We show that the requirements of energy conservation and isentropicity lead to the replacement of the usual volume calculation with a conservation integral. They further forbid the use of higher order functional representations for either velocity or stress within zones or control volumes, forcing the use of a constant stress element and a constant velocity control volume. This, in turn, causes the point and zone coordinates toformally disappear from the Cartesian formulation. The form of the work equations and the requirement for dissipation by viscous forces strongly limits the possible algebraic forms for artificial viscosity. The momentum equation and a center-of-mass definition lead directly to an angular momentum conservation law that is satisfied by the system. With a few straightforward substitutions, the Cartesian formulation can be converted to a multidimensional curvilinear one. The formulation in 2D axisymmetric geometry preserves rotational symmetry.
Space–time techniques for finite element computation of flows with moving boundaries and interfaces
We describe the space-time finite element techniques we developed for computation of fluid-structure interaction problems. The core technique is the Deforming-Spatial-Domain/Stabilized Space-Time formulation. Among the other techniques developed are, the mesh update methods and the block-iterative, quasi-direct and direct coupling methods for the solution of the fully-discretized fluid and structural mechanics equations. We present numerical examples where the fluid is governed by the Navier-Stokes equations of incompressible flows and the structure is governed by the membrane and cable equations.
Discrete Conservation Properties of Unstructured Mesh Schemes
Annual Review of Fluid Mechanics, 2011
Numerical methods with discrete conservation statements are useful because they cannot produce solutions that violate important physical constraints. A large number of numerical methods used in computational fluid dynamics (CFD) have either global or local conservation statements for some of the primary unknowns of the method. This review suggests that local conservation of primary unknowns often follows from global conservation of those quantities. Secondary conservation involves the conservation of derived quantities, such as kinetic energy, entropy, and vorticity, which are not directly unknowns of the numerical system. Secondary conservation can further improve physical fidelity of a numerical solution, but it is typically much harder to achieve. We consider current approaches to secondary conservation and techniques used outside of CFD that are potentially related. Finally, the review concludes with a discussion of how secondary conservation properties might be included automat...
On the significance of the GCL for flow computations on moving meshes
37th Aerospace Sciences Meeting and Exhibit, 1999
The objective of this paper is to establish a firm theoretical basis for the enforcement of Discrete Geometric Conservation Laws (D-GCLs) while solving flow problems with moving meshes. The GCL condition governs the geometric parameters of a given numerical solution method, and requires that these be computed so that the numerical procedure reproduces exactly a constant solution. In this paper, we show that this requirement corresponds to a time accuracy condition. More specifically, we prove that satisfying an appropriate D-GCL is a sufficient condition for a numerical scheme to be at least first order time-accurate on moving meshes.