Twenty-Third Symposium on Naval Hydrodynamics (2001)

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 897 An Unstructured Multielement Solution Algorithm for Complex Geometry Hydrodynamic Simulations D.Hyams, K.Sreenivas, C.Sheng, S.Nichols, L.Taylor, W.Briley, D.Marcum, D.Whitfield (Mississippi State University, USA) ABSTRACT The primary objective of this study is to demonstrate an efficient incompressible flow solver capable of performing time-accurate, 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 (y·~1) up to Re=109 and approximately 106 points or more. The prolate spheroid is presented as a model problem with complex flow phenomena; surface pressure distributions agree well with experimental data. Realistic applications include 1) the NOAA FRV-40 ship hull, 2) the SUBOFF hull at model scale and full scale conditions, and 3) the DTMB 5415 hull in nonappended and dynamic fully appended configurations. In all cases, agreement between computations and experimental measurements ranges from reasonable to excellent. INTRODUCTION The primary objective of this study is to develop and demonstrate an efficient RANS incompressible flow solver capable of performing time-accurate, viscous, high Reynolds number flow simulations for complex geometries using unstructured grids. This flow solver is to be demonstrated for large-scale multielement meshes with good sublayer resolution (y·~1) up to Re=109 and approximately 106 points or more with an emphasis toward hydrodynamic applications. Sample results are shown here for both surface ship and submarine geometries. The present solution algorithm is related to several previous efforts; implicit algorithms for flows on unstructured grids have been investigated extensively by a variety of authors [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]. The current approach is an evolution of the implicit flow solver and code of Anderson et al. [11] [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]. The present parallel unstructured viscous flow solver is based on a coarse-grained domain decomposition for concurrent solution within subdomains assigned to multiple processors. All tetrahedral and multielement unstructured meshes in this work are generated with an advancing normal methodology for the boundary layer elements, and an AFLR advancing front/local reconnection methodology for the isotropic elements as given in [20]. This procedure allows for the generation of high quality unstructured grids suitable for simulation of high Reynolds number viscous flows. All geometry preparation and surface grid generation is performed using SolidMesh [21] with AFLR surface grid generation [22]. This paper is organized as follows: the governing equations are outlined, followed by the numerical procedures for the discrete solution of these governing equations. Solutions of high Reynolds number flows around three complex hydrodynamic configurations and a smaller model problem are presented next to demonstrate the efficiency and accuracy of the solution technique. Conclusions are summarized in the last section. GOVERNING EQUATIONS The unsteady three-dimensional incompressible Reynolds-averaged Navier-Stokes equations are presented here in Cartesian coordinates and in conservative form. Following [23], an artificial time derivative term ( where ρd=P/β) has been added to the continuity equation to cast the complete set of governing equations into a time-marching form. The nondimen the authoritative version for attribution. 

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 899 where The matrix R is a matrix constructed from the right eigenvectors of the flux Jacobian, and Λ is a diagonal matrix whose entries contain the absolute values of the eigenvalues of the flux Jacobian. The eigensystem used in this work is based on that reported in [18]. Note that is evaluated with Roe-averaged variables, which is simply the arithmetic average between left and right solution states in the case of incompressible flows. Viscous fluxes on tetrahedral meshes are evaluated using a finite-volume approach, which is equivalent to a Galerkin finite element method. In this technique, velocity gradients are evaluated in each tetrahedral element and the viscosity is computed as the average of the nodes making up the element. With this information, the viscous flux vector is evaluated and the results scattered to the nodes comprising the element. For general element grids, it is expedient to use only edge-local information to compute the viscous fluxes. This allows the evaluation of viscous fluxes on each face of the control volume without regard to the varying element types of the mesh. An algorithm in which no element information is used outside of metric computations is termed a “grid transparent” algorithm [25]. To this end, the viscous fluxes are evaluated directly at each edge midpoint using separate approximations for the normal and tangential components of the gradient vector to construct the velocity derivatives [26]. Using a directional derivative along the edge to approximate the normal component of the gradient and the average of the nodal gradients to approximate the tangential component of the gradient leads to the following expression [27]: (10) The weighted least squares method is used to evaluate the nodal gradients in the preceding formula. Temporal Residual After the spatial terms have been suitably discretized, the time derivative term appearing in Equation 8 must be approximated. A general difference expression is available for this purpose [28] [29], and is given as follows: (11) where ∆qn=qn · 1−qn. A first order accurate in time Euler implicit scheme is given by the choices θ1=1, θ2=0. Correspondingly, a second order time accurate Euler implicit scheme is given by θ1=1, θ2=1/2. Since θ1=1 for both time discretizations used in this work, Equation 11 can be further simplified: (12) Using Equation 8 to replace the time derivative, (13) By the definition of q, one can write where is the volume averaged solution variable vector Then, the following two identities can be formed: (14) (15) Inserting the above two identities into Equation 13, one arrives at the following expression: (16) Now, one must consider the Geometric Conservation Law (GCL). This statement relates the rate of change of a physical volume to the motion of the volume faces: (17) According to Thomas and Lombard [30] and later Janus [31], the solution of the volume conservation equation must be performed in exactly the same manner as the flow equations to ensure that GCL is satisfied. This procedure ensures that spurious source terms caused by volume changes are eliminated. Using the same time differencing expression (Equation 12) to approximate Equation 17, (18) where Note that the left hand side of the preceding equation is exactly the bracketed the authoritative version for attribution. term in Equation 16. Replacing the bracketed term and rearranging slightly gives the final form of the discretization of the time derivative: (19) For incompressible flows, it is prudent to ensure a divergence-free velocity field at the end of each Newton iteration. To this end, the contribution of the time derivative and GCL terms to the residual are removed for the continuity equation only. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 900 Time Evolution A Newton iterative time evolution scheme is applied, which requires the solution of a sparse linear system at each nonlinear subiteration: (20) where (21) where Now, expanding the terms and performing the required differentiations of results in the following expression for Newton's method: (22) where For notational convenience, both the inviscid and viscous terms are collapsed into a single flux function H. Note that the iteration can be started by using an initial guess of Also, performing only one iteration of Newton's method per time step (with 1st order time discretization and no GCL terms) is equivalent to a time linearization of the spatial terms only. However, writing the method in this framework is more general than a straightforward time linearization of the nonlinear terms. To solve the resulting linear system, a bidirectional Gauss-Seidel solution algorithm is adopted. Splitting the matrix into diagonal, upper triangular, and lower triangular parts as the Gauss-Seidel sweeps may be written as the following two-step process per subiteration (k is the linear subiterative index): (23) (24) where it is understood that is evaluated at the n−1, m−1 time level and Newton subiteration. The diagonal, lower, and upper operators are defined as (25) (26) (27) The initial guess to begin the iteration is ∆Qn · 1.0=0. In the above formulas, represents the combined inviscid and viscous flux vector from node 0 to node i. N(0) represents the set of neighbors for node 0, NL(0) is the list of neighbors such that the node label ℓ(i)< ℓ(0), and NV(0) is the list of neighbors such that the node label ℓ (i)> ℓ(0) Boundary Conditions Viscous conditions are enforced by modifying the linear system such that no change is allowed in the velocity, and the pressure is driven according to the imbalance in the continuity equation in the boundary control volume [12]. Farfield conditions are handled via a characteristic variable reconstruction; all boundary conditions are handled in an implicit fashion. A symmetry plane boundary condition imposes for any arbitrary variable ψ. In addition, no flow is allowed through a symmetry surface; so, like a solid wall condition, θ=0. To simulate symmetry conditions, a layer of phantom the authoritative version for attribution. entities that is a mirror image of the entities inside and connected to the symmetry plane is created; one element layer is mirrored. Thus, control volumes on the symmetry plane are closed and behave just as interior control volumes. Phantom nodes created by the mirroring process are updated in the same hierarchy as described for phantom nodes generated by the parallelization. Note that care must be taken to copy scalars, mirror vectors, and mirror tensors appropriately to the symmetry phantom nodes such that no fluxes are allowed through the plane. Turbulence Modeling The one-equation turbulence model of Spalart and Allmaras [32] is available in the present work; this model formulates a transport equation for a working variable which is then related to the eddy viscosity: 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 901 (28) (29) The constant and secondary function definitions are given in [32]. For more accurate vortex preservation in the field, a modification to the production term is available as described in [15]. The discrete version of the transport equation is solved in an Euler implicit fashion as described for the mean flow. The two equation q−ω turbulence model is also available for simulation of turbulent effects in high Reynolds number flows. This model uses a transport equation each for the velocity scale and length scale to specify the distribution of the eddy viscosity. The velocity scale is defined as the square root of the turbulent kinetic energy The length scale ℓt is then defined by a relation between qT and ω (ℓt=qT/ω). The eddy viscosity is given by the standard definition (shown here in nondimensional form): (30) The field equation for the variables qT and ω is given in Equation 31 and Equation 32 in nondimensional form. Notice that the time rate of change of the turbulent transport variables are made up of standard contributions from convection and diffusion, plus an additional source term that models production and destruction of the variable. (31) (32) where the source terms in the field equations are defined as: (33) (34) where is the strain rate invariant and is the divergence of velocity (taken to be zero for incompressible flows). Constants are taken from the version II q−ω model given in [33]. The diffusive terms in both turbulence models are discretized in the same manner as the viscous terms for the mean flow, and the convective terms are computed via pure upwinding. Appropriate consideration is given to maintain positive operators in the formation of the Jacobian matrix for the implicit solution of the transport equation(s). The respective turbulence models are incorporated with the mean flow solution in a “loosely-coupled” procedure; that is, the core governing equations are solved first, then the turbulence model is solved independently. This procedure allows for easy interchange of the turbulence models. Figure 1: Iteration hierarchy used for the parallel unstructured solution algorithm PARALLELIZATION For quick turnaround time in a design environment, it is essential to parallelize the flow solution 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 solution algorithm employs iterative solution of the implicit approximation, with the concurrent iteration hierarchy as shown in Figure 1. Also, domain decomposition takes the authoritative version for attribution. place with each node in the domain uniquely mapped to a given task. The code employs MPI message passing for interprocessor communication. 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 one-to-one correspondence with the solution computed in serial mode. For the current numerical algorithm, this ability has been 

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 903 Figure 4: Unstructured algorithm compared to experimental data; stations x/L=0.11, 0.23, 0.31, 0.44 Figure 5: Unstructured algorithm compared to experimental data; stations x/L=0.69, 0.77, 0.83, 0.90 Figure 6: Surface grid for the fishing vessel case; experimental data is presented for r/R=0.69 NOAA FRV-40 Hull The objective of the NOAA FRV-40 flowfield study is to examine the fluid behavior in the vicinity of the propeller appendage. The propeller itself was not used in the experiment, the purpose of which was to obtain nominal propeller plane data. Experimental data is available for several circles defined around the propeller appendage; this data is compared to the computed results. The fishing vessel case utilizes approximately 937,000 points and 5.2M tetrahedra for the entire domain with a sublayer resolution of y+=2–3 on the hull; the Reynolds number is 7.4 million based on the body length. A tetrahedral grid is used exclusively for the fishing vessel case; the surface grid, as well as a location at which data is available, is shown in Figure 6. A sample of computed vs. experimental results for the fishing vessel case are shown in Figure 7 for a disk coinciding with the intended location of the propulsor; for the computed results, the one equation Spalart-Allmaras turbulence model is utilized. As shown in Figure 7, good agreement exists between computed and experimental results. A distinct velocity defect is seen below the keel (θ=0°) and behind the prop appendage (θ=180°). Also, the trends in the distributions of Vr and Vt indicate that the flow in the vicinity of the disk is directed slightly upwards. Agreement between computations and experiment improves as r/R increases, presumably due to smoothing of high gradients in the flow very near the appendage surface. A notable exception is the wake from the centerboard, which is clearly visible in the experimental results but is not predicted by the computation at large r/R. A second set of velocity profile predictions are given using the two equation q−ω turbulence model in Figure 8. the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 904 No significant differences are noted by comparing the q−ω solution with the Spalart-Allmaras turbulence model solution, (Figure 7), except that a stronger wake is predicted from the keel by the one equation model as r/R increases (thus indicating that less eddy viscosity is present in this vicinity). So, the solution itself seems to be relatively insensitive to the turbulence model used. Figure 7: Velocity profiles for the fishing vessel case at r/R=0.69; Spalart-Allmaras turbulence model SUBOFF Model For application to submarines, the SUBOFF model is presented as a candidate test case. The SUBOFF case is based on the submarine model given in [36] with flow measurements taken from [37] and [38]. The configurations tested here are the nonappended SUBOFF hull and a SUBOFF hull with four stern appendages. The Reynolds number is 14 million based on the reference length of 13.9792 feet (the body length of the hull is 14.2917 feet). The multielement unstructured mesh consists of 1.2M nodes, 1.7M prisms, and 2.3M tetrahedra. The spacing of the first mesh point from the body is 7.2x10−7, which leads to a y+ distribution of less than 0.5 over the majority of the hull body (thus indicating good viscous sublayer resolution). An overview of the SUBOFF volume grid in the vicinity of the stern appendages is given in Figure 9; this figure illustrates the high aspect ratio prismatic boundary layer elements as well as the large amount of grid point clustering in the boundary layer. A typical distribution of y+ on the submarine body is shown in Figure 10. The data given in [37] primarily consists of integrated force data given a certain model configuration and submarine attitude. The axial force, normal force, and pitching moment are computed and compared to the corresponding experimental data in Figures 11– Figure 8: Velocity profiles for the fishing vessel case at r/ R=0.69; q−ω turbulence model Figure 9: Volume grid in the vicinity of the stern appendages for the SUBOFF model the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 905 13. Agreement is excellent for the axial forces, normal forces, and pitching moments at all angles of attack, and for both Spalart-Allmaras and q−ω turbulence models. Since the experimental data itself is nonsymmetric (the drag forces for AOA=−10 should equal the drag forces for AOA=10, for example), no firm conclusion may be drawn concerning the more favorable turbulence model to select for the SUBOFF simulations. Figure 10: Computed values of y· on the SUBOFF hull at ten degrees angle of attack Figure 11: Axial force coefficient for the SUBOFF model Experimental data [38] is also available for the skin friction coefficient on the SUBOFF nonappended hull. For this case, two simulations are given: one at model scale (Re=11 million, 730,000 nodes) and one at full scale (Re=1.2 billion, 1M nodes). The q−ω turbulence model is utilized in both cases. As shown in Figure 14, agreement between experiment and computation is excellent on the aft part of the hull (where measurements Figure 12: Normal force coefficient for the SUBOFF Figure 13: Pitching moment coefficient for the SUBOFF model model the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 906 are available). Since the measurements were (obviously) taken at model scale, the numerical results for the full scale case have been subjected to a Reynolds number scaling [39] before comparison to the experimental data. Like the measurements, the computed skin friction coefficients display a definite peak where the flow accelerates (hull neck-down point, x/L≈0.77), and a decreasing Cf as the flow slows between x/L=0.8 to x/L=.95. The secondary peak at x/L=.98 is due to a small flow acceleration over the shoulder of the after-body cap. The capability to robustly simulate full scale Reynolds numbers is considered a key feature of the numerical algorithm. Figure 14: Comparison of full scale and model scale computations for the nonappended SUBOFF hull DTMB Model 5415 Hull The DTMB 5415 hull is presented here in nonappended and fully appended form (with dual propulsors). The primary purpose of the nonappended 5415 hull simulation is to examine nominal wake flow patterns. The fully appended hull with propulsors is intended to highlight the capability of the overall solution methodology to model extremely complex geometries coupled with complex unsteady flowfields. Nominal Wake Calculations Accurate prediction of the nominal wake is a key item for propulsor design. Here, a computation is presented for a DTMB 5415 nonappended hull at a Reynolds number of 12 million. The grid consists of 2.5M nodes, 2.8M prisms, and 5.9M tetrahedral elements. A rigid lid is used for the waterline, and a symmetry plane boundary condition runs lengthwise down the center of the hull model. Normal boundary spacing is set such that the nominal y+ value on viscous surfaces is 0.5. The q−ω turbulence model is used for the nominal wake calculation. Figure 15: Nominal wake calculation for DTMB 5415 nonappended hull compared to experimental measurements Axial velocity data is given in Figure 15 at the propulsor plane of x/L=0.935 [40]. The overall trends in the experimental data have been captured by the unstructured computations. The primary difference is the amount of thickening in the boundary layer along the centerline of the hull; the computations seem to overpredict this phenomenon. A probable cause of this disagreement is that the mesh used in the nominal wake calculation is insufficiently resolved to adequately capture the vortices generated by the bulbous bow; hence, the influence of the vortex structure is not reflected in the computed results. Fully Appended with Rotating Propulsors The DTMB 5415 model with appended shafts, struts, rudders, and propulsors is presented as a demonstration of capability of the current unstructured methodology to 1) model complex geometries, 2) generate high-quality viscous grids around these complex geometries, and 3) perform accurate unsteady solutions on these complicated meshes. The appended DTMB 5415 simulation is run with a Reynolds number of 12.7 million and with the Spalart-Allmaras turbulence model for closure. The grid consists of 2.15M points, 3M prisms, and 3.5M tetrahedra. A symmetry plane boundary condition is used such that only half of physical space is simulated, and a rigid lid is used for the waterline. The time step selected was 1.38x10−4, which corresponds to a prop rotation of one degree per time step with a the authoritative version for attribution. nondimensionalized rotational speed of 126.59. Local time stepping was utilized during the first part of the simulation to quickly establish wake formations, and minimum time stepping was then performed for 3.5 prop revolutions to establish periodicity. Five Newton iterations and seven linear subiterations were performed each time step. The rotation of the propulsors is handled via local grid regeneration. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 907 Figure 16: Surface pressure for the aft end of the fully appended DTMB 5415 hull with twin propulsors The pressure distribution on the hull and propulsors is shown in Figure 16. The surface pressures indicate the complexity of the flowfield, and the expected trends are apparent on the struts, shafts, rudders, and hullform. Further, the effect of the prop wash on the rudders is visible in the form of a strong low pressure region on the outboard side of the rudder and a relatively benign pressure distribution on the inboard side. Velocity data from the computations are compared to experimental measurements [41] in Figure 17. In this figure, contours of axial velocity are shown with velocity vectors shown for secondary motion. The computations have closely captured the trends reflected in the experiment, with the notable exception of the nominal wake (demonstrated in the previous section). The reason for the absence of this wake is insufficient grid resolution behind the keel, such that the convected wake structure is dissipated before it reaches the propulsor plane. Predictions of the prop wash, however, are accurate to the experimental results. Slight differences may be observed on the inboard side of the props, where the effect of the keel wake allows for easier entrainment of the flow aft of the propulsors. CONCLUSIONS A parallel unstructured solution algorithm capable of time-accurate, high Reynolds number complex geometry simulations has been presented and demonstrated on several hydrodynamic cases of interest. It is also demonstrated that rotating propulsors may be simulated effectively in an unstructured environment. Future work involves the incorporation of a nonlinear free surface capability into the unstructured solver such that propulsor/hull/free-surface interactions may be modeled and studied in detail. In addition, an effort is currently underway to incorporate general surface motion into the unstructured algorithm. This capability allows prescribed movement of the control surfaces of a given body to determine maneuvering characteristics. Figure 17: Comparisons of unstructured computation to experimental data for fully appended 5415 hull with propulsors ACKNOWLEDGEMENTS This research was sponsored by the Office of Naval Research under grant number N00014–99–1–0751 and includes work monitored by Dr. Edwin Rood and Dr. Patrick Purtell. This support is gratefully acknowledged. REFERENCES [1] Stuart E.Rogers. Comparison of implicit schemes for the incompressible Navier-Stokes equations. AIAA Journal, 33(11):2066–2072, 1995. [2] Timothy J.Barth. A 3-D upwind Euler solver for unstructured meshes. AIAA Paper 91–1548, 1991. [3] John T. Batina. Implicit upwind solution algorithms for three-dimensional unstructured meshes. AIAA Journal, 31(5), May 1993. [4] V.Venkatakrishnan and D.J.Mavriplis. Implicit solvers for unstructured meshes. Journal of Computational Physics, 105:83–91, 1993. [5] V.Venkatakrishnan. Parallel implicit unstructured grid Euler solvers. AIAA Journal, 32(10):1985–1991, October 1994. [6] Steven W.Hammond and Timothy J.Barth. Efficient massively parallel Euler solver for two-dimensional unstructured grids. AIAA Journal, 30 (4):947–952, April 1992. the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line AN UNSTRUCTURED MULTIELEMENT SOLUTION ALGORITHM FOR COMPLEX GEOMETRY HYDRODYNAMIC SIMULATIONS 908 [7] Christopher W.S.Bruner and Robert W.Walters. Parallelization of the Euler equations on unstructured grids. AIAA Paper 97–1894, 1997. [8] R.Ramamurti and R.Löhner. A parallel implicit incompressible flow solver using unstructured meshes. Computers and Fluids, 25(2):119–132, 1996. [9] Jonathan M.Weiss, Joe P.Maruszewski, and Wayne A.Smith. Implicit solution of the Navier-Stokes equations on unstructured meshes. AIAA Paper No. 97–2103, 1997. [10] D.J.Mavriplis. On convergence acceleration techniques for unstructured meshes. AIAA Paper No. 98–2966 , 1998. 29th AIAA Fluid Dynamics Conference, June 15–18, Albuquerque, NM. [11] W.Kyle Anderson. Grid generation and flow solution method for the Euler equations on unstructured grids. Technical Report L-16986, NASA Langley Research Center, Hampton, VA 23665–5225, April 1992. [12] W.Kyle Anderson and Daryl L.Bonhaus. An implicit upwind algorithm for computing turbulent flows on unstructured grids. Computers in Fluids, 23(1):1–21, 1994. [13] W.Kyle Anderson, Russ D.Rausch, and Daryl L. Bonhaus. Implicit/multigrid algorithms for incompressible turbulent flows on unstructured grids. AIAA Paper 95–1740, 1995. [14] C.Sheng, D.L.Whitfield, and W.K.Anderson. Multiblock approach for calculating incompressible fluid flows on unstructured grids. AIAA Journal, 37(2):169–176, 1999. [15] C.Sheng, D.Hyams, K.Sreenivas, A.Gaither, D.Marcum, D.Whitfield, and W.K.Anderson. Three-dimensional incompressible Navier-Stokes flow computations about complete configurations using a multiblock unstructured grid approach. AIAA Paper 99–0778, 1999. 37th AIAA Aerospace Sciences Meeting and Exhibit, January 1999, Reno, NV. [16] C.Sheng, L.K.Taylor, and D.L.Whitfield. Multiblock multigrid solution of three-dimensional incompressible turbulent flows about appended submarine configurations. AIAA Paper No. 95–0203, 1995. [17] C.Sheng, J.P.Chen, L.K.Taylor, M.Y.Jiang, and D.L.Whitfield. Unsteady multigrid method for simulating 3-D incompressible Navier-Stokes flows on dynamic relative motion grids. AIAA Paper No. 97–0446, 1997. [18] Lafayette K.Taylor. Unsteady Three-Dimensional Incompressible Algorithm based on Artificial Compressibility. PhD thesis, Mississippi State University, 1991. [19] Ramesh Pankajakshan and W.Roger Briley. Parallel solution of viscous incompressible flow on multi-block structured grids using MPI. Parallel Computational Fluid Dynamics: Implementations and Results Using Parallel Computers, pages 601–608, 1995. [20] David L.Marcum and J.Adam Gaither. Mixed element type unstructured grid generation for viscous flow applications. AIAA Paper 99–3252, 1999. 14th AIAA CFD Conference, June 1999, Norfolk, VA. [21] J.Adam Gaither. A solid modelling topology data structure for general grid generation. Master's thesis, Mississippi State University, 1997. [22] David L.Marcum. Unstructured grid generation components for complete systems. April 1996. 5th International Conference on Numerical Grid Generation in Computational Field Simulations, Starkville, MS. [23] Alexandre Joel Chorin. A numerical method for solving incompressible viscous flow problems. Journal of Computational Physics, 2:12–26, 1967. [24] P.L.Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43:357–372, 1981. [25] Andreas Haselbacher, James J.McGuirk, and Gary J.Page. Finite volume discretization aspects for viscous flows on mixed unstructured grids. AIAA Journal, 37(2):177–184, February 1999. [26] David L.Marcum. Private conversations, Engineering Research Center for Computational Field Simulation, Mississippi State, MS, December 1997. [27] D.G.Hyams, K.Sreenivas, C.Sheng, W.R.Briley, D.L.Marcum, and D.L.Whitfield. An investigation of parallel implicit solution algorithms for incompressible flows on multielement unstructured topologies. AIAA Paper 2000–0271, 2000. 38th Aerospace Sciences Meeting and Exhibit, January 2000, Reno, NV. the authoritative version for attribution. 

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