Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 897

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.

OCR for page 897

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 898
sionalized equations can be written in integral form as
(1)
where is the outward pointing unit normal to the control volume The vector of dependent variables and the
components of the inviscid and viscous flux vectors are given as
(2)
(3)
(4)
where β is the artificial compressibility parameter (typically 15 in this work), u, v, and w are the Cartesian velocity
components in the x, y, and z directions, and and are the components of the normalized control volume face
vector. Θ is the velocity normal to a control volume face:
(5)
where the grid speed Note that is the control volume face
velocity. The variables in the preceding equations are normalized with respect to a characteristic length scale (L) and
freestream values of velocity (Ux), density (ρx), and viscosity ( µx). Thus, the Reynolds number is defined as Re=UxL/vx.
Pressure is normalized with where P* is the local dimensional static pressure. The viscous stresses
given in Equation 4 are
(6)
where µ and µ i are the molecular and eddy viscosities, respectively.
NUMERICAL APPROACH
The baseline flow solver is a node-centered, finite volume, implicit scheme applied to general unstructured grids
with nonsimplical elements. The flow variables are stored at the vertices and surface integrals are evaluated on the median
dual surrounding each of these vertices. The nonoverlapping control volumes formed by the median dual completely
cover the domain, and form a mesh that is dual to the elemental grid. Thus, a one-to-one mapping exists between the
edges of the original grid and the faces of the control volumes.
The solution algorithm consists of the following basic steps: reconstruction of the solution states at the control
volume faces, evaluation of the flux integrals for each control volume, and the evolution of the solution in each control
volume in time.
Reconstruction
A higher order spatial method is constructed by extrapolating the solution at the vertices to the faces of the
surrounding control volume. The unweighted least squares method (solved via QR factorization [12]) is used to compute
the gradients at the vertices for the extrapolation. With these gradients known, the variables at the interface are computed
with a first order Taylor series as:
(7)
where is a vector that extends from node 0 to the midpoint of the edge associated with the control volume face in
question.
Residual Evaluation
The governing equations are discretized using a finite volume technique; thus, the surface integrals in Equation 1 are
approximated by a quadrature over the surface of the control volume of interest. So, the numerical discretization of the
spatial terms associated with the control volume surrounding vertex 0 results in
(8)
where the spatial residual contains all contributions from the discrete approximation to the inviscid and viscous
the authoritative version for attribution.
terms Also, the quantity q is defined as
Spatial Residual
The evaluation of the discrete residual is performed separately for the inviscid and viscous terms given in Equation
1. The Roe scheme [24] is used to evaluate the inviscid fluxes at each face of the control volume. The algebraic flux
vector is replaced by a numerical flux function, which depends on the reconstructed data on each side of the control
volume face:
(9)

OCR for page 897

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.

OCR for page 897

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)
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:

OCR for page 897

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

OCR for page 897

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 902
shown in [34]. Also, the code must be efficient in its 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. A sample scalability result is shown in Figure 2. Finally, the consequences of the
inevitable domain decomposition should not seriously compromise the convergence rate of the iterative algorithm.
Figure 2: Actual versus ideal execution times as a function of the number of processors utilized; predicted run times are from
a heuristic performance estimate described in [34].
APPLICATIONS
A key contribution in this work is to demonstrate this solution methodology on realistic hydrodynamic applications.
To this end, several complex configurations are shown. Along with the capability to handle complex geometry high
Reynolds number flows, the ability to rotate actual propulsors is demonstrated in the following results. This enabling
technology allows a true picture of propulsor-hull flow interactions in an unstructured CFD context.
Prolate Spheroid
The prolate spheroid is a surface of revolution which produces a relatively complex flowfield even in steady flow at
moderate angles of attack. Regions of laminar, transition, turbulent, and separated flow all are present; in addition, each
are functions of the angle of attack. Further, the body itself loosely resembles a submarine hull, which has obvious
relevance for a solution algorithm with a hydrodynamic focus. Experimental data is available from Meier [35] for 3D
boundary layers developing on the prolate spheroid; this data is in the form of surface pressure distributions for steady
flows at angle of attack and for unsteady pitch/plunge/turn body motions. Only the steady results are discussed here;
unsteady body motions are to be the subject of future study.
Figure 3: Pressure distribution on body surface and at x/L=0.83
The flow condition for the prolate spheroid case presented here is a Reynolds number of 4.2x106 based on the body
length. The grid utilized consists of 750,000 points, 1.2M tetrahedra, and 1M prisms. Normal grid spacing is set at 10−6,
which leads to y+ values of approximately 0.4 for the first point from the surface. Results given in this section are
computed with the second order q−ω turbulence model.
Figure 3 shows the pressure distribution on the surface of the prolate spheroid and the pressure distribution on a
cutting plane placed at x/L=0.83. The stagnation point is visible underneath the nose of the spheroid. The most striking
feature is the separation line extending the length of the body where the boundary layer is unable to remain attached to the
surface due to the high angle of attack. The separated boundary layers roll into a vortical structure on the leeward side of
the body, which causes a secondary low pressure line where the vortices impinge on the leeward surface.
A comparison of the computed results and measurements is shown in Figure 4 and Figure 5; note that a
the authoritative version for attribution.
circumferential angle of zero corresponds to the windward side of the prolate spheroid. Overall, agreement with
experimental data is very good, with the flow trends shown by the measurements being reflected in the computations. The
effect of the primary vortices on the body pressure (θ≈155°) is underpredicted somewhat by the computations at x/L=0.44
and again at x/L=0.56 (not shown); however, all other stations at which comparisons may be made demonstrate excellent
correlation between computed and measured data.

OCR for page 897

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.

OCR for page 897

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.

OCR for page 897

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.

OCR for page 897

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.

OCR for page 897

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.

OCR for page 897

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.

OCR for page 897

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 909
[28] R.M.Beam and R.F.Warming. An implicit factored scheme for the compressible Navier-Stokes equations. AIAA Journal, 16(4):393–402, April 1978.
[29] Lafayette K.Taylor, J.A.Busby, M.Y.Jiang, A.Arabshahi, K.Sreenivas, and D.L.Whitfield. Time accurate incompressible Navier-Stokes simulation
of the flapping foil experiment. In Sixth International Conference on Numerical Ship Hydrodynamics, August 1993. Iowa City, Iowa.
[30] P.D.Thomas and C.K.Lombard. Geometric conservation law and its application to flow computations on moving grids. AIAA Journal, 17(10):1030–
1037, 1978.
[31] J.Mark Janus. Advanced 3-D CFD Algorithm for Turbomachinery. PhD thesis, Mississippi State University, May 1989.
[32] P.R.Spalart and S.R.Allmaras. A one-equation turbulence model for aerodynamic flows. AIAA Paper 92–0439, 1992.
[33] T.J.Coakley and T.Hsieh. A comparison between implicit and hybrid methods for the calculation of steady and unsteady inlet flows. AIAA Paper
85– 1125, 1985. AIAA/SAE/ASME/ASEE 21st Joint Propulsion Conference, July 1985, Monterey, California.
[34] Daniel G.Hyams. An Investigation of Parallel Implicit Solution Algorithms for Incompressible Flows on Unstructured Topologies. PhD thesis,
Mississippi State University, May 2000.
[35] H.U.Meier, H.P.Kreplin, and H.Vollmers. Development of boundary layers and separation patterns on a body of revolution at incidence. Second
Symposium on Numerical and Physical Aspects of Aerodynamic Flows , 1983. California State University, Long Beach, California.
[36] Nancy C.Groves, Thomas T.Huang, and Ming S. Chang. Geometric characteristics of DARPA SUBOFF models. Technical Report DTRC/
SHD-1298–01, David Taylor Research Center, Bethesda, Maryland 20084–5000, March 1989.
[37] Robert F.Roddy. Investigation of the stability and control characteristics of several configurations of the DARPA SUBOFF model from captive-
model experiments. Technical Report DTRC/SHD-1298–08, David Taylor Research Center, Bethesda, Maryland 20084–5000, September 1990.
[38] T.T.Huang, N.C.Groves, T.J.Forlini, J.N.Blanton, and S.Gowing. Measurement of flows over an axisymmetric body with various appendages.
Nineteenth Symposium on Naval Hydrodynamics, August 1992. Seoul, Korea.
[39] H.Schlichting. Boundary Layer Theory. McGraw-Hill Book Company, Seventh edition, 1979.
[40] T.Ratcliff. Private conversations, Taylor Model Basin, Carderock Division, W. Bethesda, MD, May 1998.
[41] T.Ratcliff. Private conversations, Taylor Model Basin, Carderock Division, W. Bethesda, MD, July 1999.
the authoritative version for attribution.