Daniel Hyams

An investigation of parallel implicit solution algorithms for incompressible flows on multielement unstructured topologies

The primary objective of this study is to develop an efficient incompressible flow solver capable of performing viscous, high Reynolds number flow simulations for complex geometries using general unstructured grids. This parallel flow solver is demonstrated for large-scale meshes with viscous sublayer resolution (p+ N 1) and approximately lo6 points or more. Primary issues addressed in this work are 1) treatment of the connectivity between subdomain interfaces, 2) proper definition of the iteration hierarchy, and 3) methods for coupling of subdomains. The present parallel unstructured viscous flow solver is based on a domain decomposition for concurrent solution within subdomains assigned to multiple processors. The solution algorithm employs iterative solution of the implicit approximation, with coupling between subdomains according to several schemes that are a primary focus of the study. MPI message passing is used for interprocessor communication. Applications include 1) a full-scale ship hull, 2) the SUBOFF model hull with stern appendages, and 3) a fully-configured high-lift transport. Introduction Implicit algorithms for flows on unstructured grids have been investigated extensively by a variety of authors [l] [2] [3] [4]. However, implicit algorithms are much more difficult to) parallelize, because of their inherent global dependencies. As such, the parallelization of unstructured Euler solvers [5] [S] [7] and Navier-Stokes solvers [S] [9] [lo] have been previously investigated. This work seeks to examine a relaxation-type algorithm *Research Assistant I, Member ASME *Research Engineer I, Member AIAA SProfessor, Member AIAA §Distinguished Professor, Member AIAA Copyright@2000 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. in depth to provide insight concerning issues that arise in the parallelization of implicit solution algorithms on unstructured topologies. In general, the parallelization of an existing validated flow solver should satisfy several constraints. First and most important; the accuracy of the overall numerical scheme must not be compromised; i.e., the solution computed in parallel must have a oneto-one correspondence with the solution computed in serial mode. Also, the code must be efficient irrits use of computational resources. This characteristic is measured in terms of memory usage and scalability, as well as the fact that the parallel code should degenerate to the serial version if only one processor is available. Finally, the consequences of the inevitable domain decomposition should not seriously compromise the convergence rate of the iterative,algorithm. The present parallel unstructured viscous flow solver is based on a coarse-grained domain decomposition for concurrent solution within subdomains assigned to multiple processors. The solver also has the capability to map an arbitrary number of subdomains to a physical processor; thus, some flexibility is available to leverage available memory should memory resources be scarce. The present solution algorithm is related to several previous efforts. The approach is an evolution of the implicit flow solver and code of Anderson et al. [ll] [12] [13]; the solver developed in this series of works demonstrates 3D, implicit, high Reynolds number solution capability. Also, this work follows the unstructured multiblock solver of Sheng and Whitfield [14] [15] which uses the same core solver but employs a multiblock technique to reduce memory consumption by 70%. These studies are in turn related to the multiblock structured solvers originating from Taylor, Whitfield, and Sheng [16] [17] [18]. Elements of the present approach to parallel solution are related to the parallel multiblock structured grid solver of Pankajakshan and Briley [19].

A Generalized, Unstructured Interpolative Interface Method for Rotor-Stator Interactions

A generalized interpolative interface is developed to provide interdomain and intradomain coupling between multielement unstructured grids, including those that require highly stretched anisotropic meshes. The method centers around an extruded interpolative interface that does not require matched unstructured grids on the corresponding surfaces. In the context of rotor/stator interactions, this generalized interpolative interface is utilized between the rotor and stator sections, so that each can be solved within its own frame of reference. The interpolative interface supports the sliding of the rotor relative to the stator, such that variables are transmitted from one domain to the other accordingly. Coupling between the rotor and stator sections is accomplished via solution interpolations from mesh extrusions constructed from the interpolative interface. Further, the same interpolative interface technology is utilized to implement an axisymmetric boundary condition that also does not require matching surface grids on the periodic surfaces. Given that these axisymmetric boundary conditions intersect with the rotor/stator interface, special consideration is required in these areas, and the techniques applied are explained in this work in detail. This interpolative interface scheme is tested and applied via a performance mapping of the SDT2-R4 rotor/stator configuration as documented in Hughes, 1 simulated as a 2 rotor/5 stator passage. Results for this configuration are compared to available experimental data as well as full-wheel simulations of the same, in order to determine the eect of the axisymmetric interfaces on the overall solution. Overall agreement with experimental performance mappings is excellent.

Unstructured Nonlinear Free Surface Flow Solutions: Validation and Verification

A nonlinear free surface solution methodology is incorporated within an existing three-dimensional, unstructured Navier-Stokes solver, . This portable, parallel solver uses a node-based finite volume method to solve the incompressible Reynolds-averaged NavierStokes equations on mixed element high-aspect ratio grids, includes several turbulence models to simulate the affects of turbulence within the boundary layer, and has the capability to simulate flow through rotating propellers accurately. To obtain a nonlinear free surface, the kinematic free surface equation is solved at each time level via a finite element implementation, valid on both triangles and quadrilaterals; after several time steps (approximately 200), the grid is moved to match the free surface elevations while conforming to the geometry. Robust grid movement is achieved by using a three-dimensional extension of Farhat’s torsional spring analogy. Validation studies include a grid refinement for the free surface on a circle based on a prescribed velocity field, and of flow around a submerged NACA0012 hydrofoil. For the NACA0012 hydrofoil, for the viscous and inviscid Wigley hull and for the DTMB Model 5415 series hull, wave profiles along the hull are compared against available experimental results as well as numerical results from a more mature structured nonlinear free surface code UNCLE.

Three-Dimensional Incompressible Navier-Stokes Flow Computations about Complete Configurations Using a Multiblock Unstructured Grid Approach

A multiblock unstructured grid approach is presented for solving three-dimensional incompressible inviscid and viscous turbulent flows about complete configurations. The artificial compressibility form of the governing equations is solved by a node-based, finite volume implicit scheme which uses a backward Euler time discretization. Point Gauss-Seidel relaxations are used to solve the linear system of equations at each time step. This work employs a multiblock strategy to the solution procedure, which greatly improves the efficiency of the algorithm by significantly reducing the memory requirements by a factor of 5 over the single-grid algorithm while maintaining a similar convergence behavior. The numerical accuracy of solutions is assessed by comparing with the experimental data for a submarine with stem appendages and a high-lift configuration.

A primitive-variable Riemann method for solution of the shallow water equations with wetting and drying

A Riemann flux that uses primitive variables rather than conserved variables is developed for the shallow water equations with nonuniform bathymetry. This primitive-variable flux is both conservative and well behaved at zero depth. The unstructured finite-volume discretization used is suitable for highly nonuniform grids that provide resolution of complex geometries and localized flow structures. A source-term discretization is derived for nonuniform bottom that balances the discrete flux integral both for still water and in dry regions. This primitive-variable formulation is uniformly valid in wet and dry regions with embedded wetting and drying fronts. A fully nonlinear implicit scheme and both nonlinear and time-linearized explicit schemes are developed for the time integration. The implicit scheme is solved by a parallel Newton-iterative algorithm with numerically computed flux Jacobians. A concise treatment of characteristic-variable boundary conditions with source terms is also given. Computed results obtained for the one-dimensional dam break on wet and dry beds and for normal-mode oscillations in a circular parabolic basin are in very close agreement with the analytical solutions. Other results for a forced breaking wave with friction interacting with a sloped bottom demonstrate a complex wave motion with wetting, drying and multiple interacting wave fronts. Finally, a highly nonuniform, coastline-conforming unstructured grid is used to demonstrate an unsteady simulation that models an artificial coastal flooding due to a forced wave entering the Gulf of Mexico.

A Generalized Interpolative Interface for Parallel Unstructured Field Solvers

A generalized interpolative interface is developed to provide a coupling between multielement unstructured grids potentially in relative motion, including those that require highly stretched anisotropic meshes. The method centers around an extruded interpolative interface that does not require matched unstructured grids on the corresponding surfaces. The method is also constructed such that is generally applicable to any data that must be transported between the adjacent solution domains; as such, the same mechanism can be used regardless of the underlying governing equations of the field. This feature is utilized in the current work by using dierent sets of governing equations for the presented solutions, as well as using the same interpolative procedure in the process of solving the turbulence models. Mesh extrusions are constructed from the interpolative interface, which allow for closure and the solution of control volumes that lie on the interface. For each extruded point, a corresponding virtual point is created in order to control the exact location at which the client data interpolations are performed. This allows interpolation from locations that are reflective of the physics of the problem. Special procedures, such as utilization of surface projections and parallel boundary layer displacement algorithms, are required for support of highly stretched anisotopic grids commonly used in the resolution of large gradients most common in fluid flow solvers. All algorithms used to extrude the interpolative surface, place virtual points, and interpolate for the client data must be parallelized for compatibility with modern parallel field solvers. To this end, fully general parallel mechanisms are implemented in order to transport data from its native storage to a possibly remote location. Interpolation also requires a parallel unstructured multielement search algorithm, which is a concerted eort by itself, and is the subject of an upcoming paper. This interpolative interface scheme is validated on a ramp immersed in supersonic flow, where the shock passes through the interface; comparisons with the theoretical solution for an oblique shock are excellent. Also, an unsteady pitching airfoil in which the airfoil is pitched with the aid of a surrounding interpolative interface is examined and compared against experimental data and a baseline case where the entire grid is pitched; comparisons here are excellent as well. The NASA SDT2-R4 turbofan stage is also presented as a demonstration of capability of the interpolative interface in real-world problems. Agreement with experimental data for demonstrated case at 100% speed is good.

Computation of Vortex-Intensive Incompressible Flow Fields

The objective of this study is to use UNCLE, an unstructured, parallel, Reynolds averaged Navier-Stokes solver, to compute vortex intensive flow fields and compare the results with experimental data. Vortex intensive flow fields arise under a variety of conditions. The ones considered in this study are those arising due to rotating propellers and due to flow past submarine hulls. The examples considered here involve computations at design as well as offdesign conditions for the propellers and the SUBOFF model at various angles of drift. Results are presented for propeller P-4381 (overall thrust and torque coefficients) as well as for propeller P-5168 (velocity profiles downstream of the propeller). Forces and moments are also compared for the SUBOFF hull at various angles of drift to experimental data. Good agreement with experimental data is obtained for the various cases.

Effect of Casing and Tip Modifications on the Performance of an Axial Flow Stage

This paper is a presentation of a computational study focused on modifications to the rotor blade tip and casing end wall for the purpose of enhancing the performance and increasing the stall margin of a model turbofan stage. Grooves in the casing outboard of the rotor blade tips, as well as a very small ring protruding inward from the casing just upstream of the leading edge and downstream of the trailing edge of the rotor blade constitute the modifications that will be examined. The results will be compared to experiment as well as simulations of the baseline configuration in order to demonstrate the degree of effectiveness of the respective modifications. It is felt that this work serves as a good starting point for further investigations into possible improvements of performance and stability for both fan and compressor stages.

A Parallel Universal Mesh Deformation Scheme

Many approaches for moving and deforming mesh have been developed, but the approach adopted often depends on both the meshing scheme and the proposed application. Approaches based on a spring analogy with linear torsional springs or solution of partial differential equations have been used, but are generally very expensive to solve at each time step and are not trivial to parallelize. Here, a universal approach to g rid motion known as the algebraic interpolation method (AIM) is followed to manage deforming surfaces. This method is universal and applicable to any grid type. Also , it is perfectly suitable to a parallel platform and can be implemented efficiently. The original scheme has some difficulty handling two-node bending mesh deformation involved in various fluid -structure interaction problems and other cases in which mesh deformation is driven solely by the surface motion. Several modifications have been made for these applications. It is determined that the grid quality can be improved significantly by adding a smoothing algorithm. Extra connectivities can also help improve the grid quality. The current scheme is applied to several well known synthetic jet applications from a NASA Langley Workshop for validation. Results are presented for the mesh deformation of NACA0012 airfoil and Suboff body. Free surface evolution of S175 container ship is also included along with its two -node bending mesh deformation.

Arbitrary Overlapping Interfaces for Unsteady Unstructured Parallel Flow Simulations

A sliding interface technique is presented in order to simulate the flowfield for geometries involving relative motion using a parallel unstructured flow solver. The moving subdomain grids are discontinuous at the subdomain boundary but are coupled through the sliding interface. The sliding interface is created by extruding the faces into the adjacent subdomain in order to compute a flux across the interface. The values for the extruded nodes are found by interpolation. Multiple cases were examined to determine the effect of the interface on the flowfield. Even though explicit flux conservation is not enforced across the subdomain interface, good results are still obtained.