ANSYS CFX–10 RANS Normal Force Predictions for the ...ANSYS CFX–10 RANS Normal Force Predictions for the Series 58 Model 4621 Unappended Axisymmetric Submarine Hull in Translation

Defence R&D Canada – Atlantic

DEFENCE DÉFENSE&

ANSYS CFX–10 RANS Normal Force Predictions

for the Series 58 Model 4621 Unappended

Axisymmetric Submarine Hull in Translation

George D. Watt

DRDC Atlantic

Christopher R. Baker and Andrew G. Gerber

University of New Brunswick

Technical Memorandum

DRDC Atlantic TM 2006-037

September 2006

Copy No.________

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

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ANSYS CFX–10 RANS Normal Force Predictions

for the Series 58 Model 4621 Unappended

Axisymmetric Submarine Hull in Translation

George D. Watt, DRDC AtlanticChristopher R. Baker, University of New BrunswickAndrew G. Gerber, University of New Brunswick

Defence R& D Canada – AtlanticTechnical MemorandumDRDC Atlantic TM 2006-037September 2006

Abstract

The ability of the commercial Reynolds Averaged Navier Stokes (RANS) software packageANSYS CFX–10 to predict the normal force on an unappended submarine hull at uniformincidence angles up to 18 degrees is evaluated. Hybrid meshes of several million nodes each arecreated using a script to adapt the mesh to the body vortex for various angles of attack. Aformal verification procedure was attempted and shown to be dependent on solutions convergedto four or five significant figures. Numerical discretization error is shown to be much smallerthan the differences between experimental and RANS normal forces. Turbulence modelling hasa marked effect on these differences, which are greater than 20% for the Shear Stress Transporttwo-equation model and 10% for the Baseline Reynolds Stress seven-equation model.

Resume

Les capacites predictives du logiciel d’analyse d’equations de Navier-Stokes a moyenne deReynolds (RANS) ANSYS CFX–10 pour la force normale exercee sur une coque de sous-marin sans appendices a des angles d’incidence uniformes allant jusqu’a 18 degres sont encours d’evaluation. Des mailles hybrides, chacune comportant plusieurs millions de nodes, sontcreees a l’aide d’un scenario pour adapter la maille aux turbulences autour d’une forme, adivers angles d’incidence. On a tente une procedure de verification normative et on a constatequ’elle dependait des solutions convergentes a quatre ou cinq chiffres significatifs. L’erreur dediscretisation numerique se revele etre beaucoup plus petite que les differences entre les forcesexperimentales et les forces normales des equations RANS. La modelisation des turbulencesproduit un effet notable sur ces differences, qui sont superieures a 20% pour le modele a deuxequations des contraintes de cisaillement au transport et de 10% pour le modele a sept equationsdes contraintes de base de Reynolds.

DRDC Atlantic TM 2006–037 i

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ii DRDC Atlantic TM 2006–037

Executive Summary

Introduction

Reynolds Averaged Navier Stokes (RANS) predictions of the real flows about ships and subma-rines are among the best available to the computational fluid dynamicist. However, they aredifficult to validate and not always accurate, especially for extreme maneuvers that often deter-mine operational limits. This study is one of three in which RANS predictions of the flow aboutthe simplest of geometries, the unappended axisymmetric submarine hull, in translation at in-cidence are compared with good quality model scale data. A previous study evaluated RANSpredictions of the DRDC standard axisymmetric hull, this study uses the Series 58 Model 4621hull, and a follow-on study will use the Defense Advanced Research Projects Agency (DARPA)SUBOFF hull.

The Model 4621 study is a US/UK/Canadian collaboration which allows for a more exten-sive evaluation of modelling options than occurred previously.

Significance

Although the unappended axisymmetric hull is a simple shape, predictions of the flows aboutit are less accurate than for submarine configurations with appendages. This is because thegeometry has no sharp edges (eg, the trailing edges of appendages) that fix the location of flowseparation. Flow separation occurs on a hull at even low incidence angles. The forces thatresult are dependent on the location of the separation and on the prediction of viscous lossesin the separated wake. Predicting these flows well is an important first step in validating ourRANS capability.

Principal Results

As was the case with the earlier DRDC standard hull calculations, appreciable differencesoccurred between the RANS predictions and experiment. However, this study carried outadditional calculations that allowed the accuracy of the numerical predictions to be assessedindependently (verification) of the RANS modelling assessment (validation). The latter resultsshow that computationally efficient turbulence models underpredict the normal force by morethan 20% and a computationally intensive turbulence model underpredicts the normal forceby 10%.

Further Work

RANS predictions for the DARPA SUBOFF hull will be evaluated next. This work will be car-ried out in collaboration with several other countries, all of whom participate in the SubmarineHydrodynamics Working Group.

G.D. Watt, C.R. Baker, and A.G. Gerber, 2006. ANSYS CFX–10 RANS Normal Force Predic-tions for the Series 58 Model 4621 Unappended Axisymmetric Submarine Hull in Translation.DRDC Atlantic TM 2006–037. Defence R& D Canada – Atlantic.

DRDC Atlantic TM 2006–037 iii

Sommaire

Introduction

Les predictions des equations de Navier-Stokes a moyenne de Reynolds (RANS) sur les ecoule-ments reels sur les navires et les sous-marins sont parmi les meilleures existantes pour le dy-namicien des fluides numeriques. Elles sont cependant difficiles a valider et ne sont pas tou-jours precises, surtout dans le cas de manœuvres extremes qui determinent souvent des limitesoperationnelles. La presente etude est l’une de trois ou les predictions des equations RANS surles ecoulements autours d’une des formes geometriques les plus simples, la coque axisymetriquede sous-marin sans appendices, qui se deplace a un angle d’incidence, sont comparees a desdonnees d’un modele a echelle de bonne qualite. Une etude precedente evaluait les predictionsdes equations RANS sur une coque axisymetrique normale du RDDC; cette etude se base surla coque de la serie 58, modele 4621, et une etude subsequente se fondera sur la coque DARPASUBOFF de l’Advanced Research Projects Agency des Etats-Unis.

L’etude du modele 4621 resulte d’une collaboration entre les E.-U., le Royaume-Uni et leCanada; elle permet une evaluation plus poussee des possibilites de modelisation que sur lesetudes precedentes.

Portee

Bien que la coque axisymetrique sans appendices soit une forme simple, les predictions sur lesecoulements autour de cette coque sont moins precises que dans le cas de configurations desous-marins avec appendices. Cette situation s’explique par le fait que cette forme geometriquene comporte aucun rebords nets (p. ex., les bords de fuite d’appendices) qui fixent les endroitsde separation des ecoulements. Ces phenomenes se produisent sur une coque meme a desangles d’incidence faibles. Les forces qui en resultent dependent de l’endroit de la separationet sur la prediction des pertes visqueuses dans un sillage separe. Une prediction precise deces ecoulements constitue la premiere etape importante pour la validation de nos capacitesd’analyse par equations RANS.

Principaux resultats

Tout comme dans le cas des calculs precedents de RDDC sur une coque normale, des differencesappreciables ont ete constatees entres les predictions par les equations RANS et les experiences.Toutefois, cette etude a procede a des calculs supplementaires qui ont permis d’evaluer demaniere independante la precision des predictions numeriques (verification) de l’evaluation dela modelisation par equations RANS (validation). Ces derniers resultats demontrent que desmodeles numerises de turbulences a haut rendement sous-estiment la force normale par plus de20%, et qu’un modele numerise de turbulences intenses sous-estiment la force normale par plusde 10%.

Recherches futures

Les predictions des equations RANS pour la coque DARPA SUBOFF feront l’objet d’uneevaluation subsequente. Ces travaux seront executes en collaboration avec plusieurs autrespays, qui participent tous au groupe de travail sur l’hydrodynamique des sous-marins.

G.D. Watt, C.R. Baker et A.G. Gerber, 2006. ANSYS CFX-10 Predictions RANS des forcesnormales pour la coque de sous-marin sans appendices axisymetrique de la serie 58, modele 4621en mouvement. RDDC Atlantique TM 2006-037. R&D pour la defence Canada – Atlantique.

iv DRDC Atlantic TM 2006–037

Table of Contents

Abstract . . . . . . . . . . . . . . . . i

Executive Summary . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . v

Nomenclature . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . 1

2 Scripted Mesh . . . . . . . . . . . . . . . 3

3 RANS Simulation . . . . . . . . . . . . . . 8

4 Overall Force and Moment Results . . . . . . . . . . 9

5 Normal Force Distributions . . . . . . . . . . . . 14

6 Concluding Remarks . . . . . . . . . . . . . . 16

References . . . . . . . . . . . . . . . . 17

DRDC Atlantic TM 2006–037 v

Nomenclature

z, w, Z

y, v, Y

x, u, X

CB

N

M

K

CB Center of buoyancy.

d Maximum hull diameter.

K,M,N Body axis moments.

K ′,M ′, N ′ Body axis moments nondimensionalized by ρU2ℓ3/2.

ℓ Overall length of the hull.

p Pressure.

r Geometric progression expansion rate of mesh spacing; the ratio ofadjacent spacings is 1 + r.

r Ratio of average coarse mesh spacing to average fine mesh spacingwhen comparing scaled meshes.

R Reynolds number, based on ℓ .

u, v,w Body axis velocities.

U =√

u2 + v2 + w2 Overall speed of vehicle.

x, y, z Body fixed axes.

X,Y,Z Body axis forces.

X ′, Y ′, Z ′ Body axis forces nondimensionalized by ρU2ℓ2/2.

y+ Dimensionless distance from the wall to the first node.

α = tan−1 w

uAngle of attack.

ρ Fluid density.

ξ = −x − xnose

ℓDimensionless distance from nose in the −x direction.

σ Uncertainty; one standard deviation of error.

vi DRDC Atlantic TM 2006–037

1 Introduction

DRDC and the University of New Brunswick (UNB), the UK Ministry of Defence and QinetiQ,and the Naval Surface Warfare Center, Carderock Division (NSWCCD) are collaborating toevaluate the ability of Reynolds Averaged Navier-Stokes (RANS) codes to predict the normalforce along an axisymmetric hullform at constant incidence to a uniform onset flow. Thisreport is concerned with evaluating ANSYS CFX–10 RANS force predictions for the US Series58 Model 4621 hullform.

Previous calculations by UNB with ANSYS CFX1 on the DRDC Static Test Rig (STR)hullform [1] underpredicted normal force and energy loss in the cores of the body vortices. Thiswork suggested that discretization of the RANS model needed improvement. US predictionsfor the 4621 hullform using a variety of other RANS codes showed a similar tendency to un-derpredict normal force, and that the underprediction was sensitive to the turbulence modelused. Best results were achieved with recent versions of the k-ω and Reynolds Stress turbulencemodels. The US in-house code IFLOW was very successful in predicting the normal force onModel 4621 [2] using the Wilcox k-ω turbulence model.

Since QinetiQ and DRDC both use ANSYS CFX, it was decided a collaborative effort tovalidate this code against the US supplied Model 4621 experimental data would be useful. The4621 shape is compared with other axisymmetric hullforms of interest in Figure 1. The hullis fatter than most and is well streamlined without a parallel midsection. Full details of the4621 geometry are available in Randwijck and Feldman [3] who also describe the experimentsin which the overall normal force, pitching moment, and axial distribution of normal force weremeasured on this model. Sung [2] also presents these data, acquired at NSWCCD at a Reynoldsnumber of 11.7 million.

0 1−0.1

0.0

0.1

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..................................................................................................

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4621, ℓ/d = 7.34

SUBOFF, ℓ/d = 8.58

STR, ℓ/d = 8.75

Figure 1 The Series 58 Model 4621 hull (solid black line, top and bottom), the Defense Ad-vanced Research Projects Agency (DARPA) SUBOFF hull (top red dashed line), and the DRDCStatic Test Rig (STR) hull (bottom blue dashed line). Plotted to scale.

The 4621/ANSYS CFX validation is carried out by UNB, QinetiQ, and DRDC using thesame hybrid mesh adapted for each angle of attack. Mesh adaption is achieved using a scriptedalgorithm (Watt et al [4]) summarized in the next section. The algorithm attempts to minimizediscretization error through the separating shear layer and body vortex on a half-model of thehull. The meshes it produces for this study have from 3 to 9 million nodes and nominal y+

spacing to the first node off the hull surface varying from 0.2 at the stern to 0.04 at the nose.

The meshing script provides for simple and accurate mesh scaling which facilitates for-mal verification of the numerical solutions. This analysis, and its current shortcomings, are

1 www.ansys.com/cfx

DRDC Atlantic TM 2006–037 1

NSWCCD Experimental data from straightline towing tank tests at R = 11.7 million atα = −18 to 18 degrees in 1 to 2 degree increments [2,3].

UNB Scripted mesh development.

Shear Stress Transport (SST) turbulence model calculations at α = 0, 6, 12,18 degrees.

SST, α = 18 degree calculations with the mesh scaled by factors of 0.5 and2.0.

SST, α = 18 degree calculation with the less accurate ‘Hi-Resolution’ dis-cretization option.

QinetiQ Baseline Reynolds Stress (RSM-BSL) turbulence model calculations at α = 0,6, 12, 18 degrees.

RSM-BSL, α = 18 degree calculations with the mesh scaled by factors of 0.5and 2.0.

RSM-BSL, α = 18 degree calculation with the less accurate ‘Hi-Resolution’discretization option.

DRDC Scripted mesh development.

α = 18 degree calculations with the following turbulence models [5]:

• Spalart-Allmaras, a popular one-equation model (CFX–10 beta feature).

• k-ǫ with scalable wall functions.

• RNG k-ǫ with scalable wall functions.

• Wilcox k-ω .

• Baseline (BSL) k-ω (blending of k-ω near a wall with k-ǫ in the outerflow).

• SST, the BSL k-ω model with improved Shear Stress Transport mod-elling.

Table 1 Contributions to the 4621/ANSYS CFX validation.

presented below. Mesh scaling, plus the ability of the script to methodically and reproducablyadapt the mesh to the flow, allows discretization error to be minimized, quantified, and assessedseparately from modelling error. Discretization error is shown to be much less than the differ-ence between the RANS predictions presented herein and experiment. Hence, this difference isprimarily attributable to RANS modelling error, an important conclusion of this study.

The most obvious source of RANS modelling error is the turbulence model. In anticipationof this, the validation considered several turbulence models. The main models assessed arethe Shear Stress Transport (SST) 2-equation and the Baseline Reynolds Stress (RSM-BSL)7-equation models, as these are the recommended models to use with CFX for this kind ofexternal flow [5]. Additional models are available and are also examined, but in less detail.Table 1 lists the effort involved.

2 DRDC Atlantic TM 2006–037

RANS calculations are most robust when a first order spatial discretization scheme is used.But first order discretization suffers from excessive numerical diffusion [6]. The diffusion prob-lem can be alleviated by using a second order scheme, but then convergence is more dependenton having a good mesh. Most commercial RANS codes use a blend of first and second orderdiscretization to gain benefits from each. The default ‘Hi-Resolution’ method in ANSYS CFXuses such a blend. For the present calculations, blending is turned off and a pure second orderdiscretization method employed. With the current meshing algorithm, there was little difficultyconverging the calculations. To see the effect of using lower order discretization, ‘Hi-Resolution’blending was used with the SST and RSM-BSL turbulence models at a single incidence angle.

2 The Scripted Mesh

The mesh building algorithm is written in the open source Tcl scripting language,1 customizedto interface with the Gridgen2 hybrid mesh generator. The algorithm is described by Wattet al [4] in some detail and only a general overview is provided here. The mesh used forthe 18 degree incidence calculations is shown in Figures 2 and 3. It consists of a fine innerstructured mesh and a coarser outer structured mesh with unstructured elements transitioningbetween the two. All cells in this hybrid mesh are connected in a one-to-one manner so thatthe discretization algorithm maintains second order accuracy throughout the flow field.

The half-model mesh is built in hull body axes but with the nose translated to the origin.The x axis points forward in-line with the hull centerline, the symmetry plane is the xz plane,and the y axis points laterally out of the page in Figures 2b and 2c. The far field boundaries arerotated to align with the far field flow and translated slightly to accommodate the unstructuredtransition region. The hull remains nominally centered within the boundaries. Building themesh in body axes simplifies postprocessing. Aligning the far field boundaries with the flowsimplifies the application of boundary conditions and, in the case of the sides, improves theaccuracy of the opening boundary condition (discussed in the next section).

Some of the script parameter settings for the Figure 2 mesh are listed next. When a settingis given two values in braces, as in {1.0 0.5}, the first value applies at α = 0 and the last atα = 30 degrees, with linear interpolation used for intermediate α settings. This is a simplescripting mechanism that allows parameters to vary with incidence.

Geometry and Flow Conditions

• Hull length ℓ = 4.572 m: sets the size of the mesh.

• α = 18 degrees: sets the angle of the far field structured mesh relative to the hull centerlineand is used in determining values for incidence dependent parameters.

• Reynolds number = 11.7 million: used with the y+ setting to estimate the normal spacingfrom the wall.

Mesh Scaling

• Mesh Scaling = 1.0: scale input spacings by this amount, as described in the Appendix ofReference [4].

1 www.tcl.tk2 www.pointwise.com/gridgen


a) Looking forward at the inlet and the topside of the opening. b) Elevation view of symmetry plane.

c) Hull structured mesh and trailing wake.

Figure 2. The Model 4621 mesh at 18 degrees incidence has 4.8 million nodes and 7.3 millionelements.

Radial Spacings

• y+ = 0.2: maximum y+ value in the mesh, at the stern on the leeward symmetry plane.

• y+ axial funneling = 0.2: moving forward from the stern into the vortex inception region,reduce the y+ spacing by this factor.

• y+ nose compression = 0.5: moving forward from the vortex inception region onto thenose, reduce the y+ spacing by this amount.

• y+ windward compression = {1.0 0.5}: moving from the leeward symmetry plane to theseparation line, reduce the y+ spacing by this factor.


0

R

R + Rlee

2

Rlee

Φ

Ylee

a) Required empirical parameters Φ, Rlee, Ylee.

stage 1

stage 2

stage 3

HDL

stage 3

aL

aW

aS

leewarddomain

windwarddomain

b) Station spacing. (Coarse mesh.)

Figure 3. The hull structured mesh is adapted to the body vortex using the parameters shown.These are specified independently at several axial stations and include the local separation point,body vortex size parameters, and radial and circumferential spacing parameters. Axial spacingparameters (not shown) must also be specified.

• Nominal aspect ratio of cells in the vortex core = 1.0: this sets the constant stage 2 radialspacing (see Figure 3) on the leeward symmetry plane based on the aL circumferentialspacing described below.

• High Density Layer (midbody) = 0.1 hull radii: stage 1 and 2 circumferential grid linesare compressed into this radial thickness on the windward domain.

• HDL (stern) = 0.5 hull radii: stage 1 and 2 circumferential grid lines are compressed intothis radial thickness at the stern.

• Geometric progression radial expansion parameter r = 0.15: used through stages 1 and 3(see the Appendix of [4] for a description of geometric progression in the current context).

Circumferential Spacings

All spacings are in degrees. For a given axial location aft of vortex inception, circumferentialspacing aW is constant from the windward symmetry plane until just before the separationline when it starts gradually changing to match aS the separation line circumferential spacing.Spacing varies smoothly from the separation line to aL at the leeward symmetry plane unless


the ‘bulge’ factor is other than 1.0, in which case that factor gives the relative increase in nodaldensity between the specified aS and aL spacings.

• Windward spacing aW = {2.5 2.0}.

• Separation line spacing aS = {2.5 1.5}.

• Leeward symmetry plane spacing aL = {2.5 2.0}.

• Bulge factor = {1.0 1.5}.

• Expansion from aS to aW spacing r = 0.1.

Axial Spacings

A butterfly mesh is used at the nose with nominal cell aspect ratios of 1.0. Moving aft, spacingexpands to a maximum and then contracts at the tail tip. At the tip, rapid contraction takesplace so the last axial space just before the truncated tail tip equals the y+ spacing. Thisprovides good convergence at this backwards facing step (Figure 4).

• Spacing over the nose tip = 0.005 hull diameters.

• Maximum mid-hull spacing = 0.01ℓ.

• Geometric progression expansion rates r = 0.05.

• Expansion forward from y+ spacing at trailing tip r = 0.5.

a) Tail truncation = 4 × 10−5 ℓ b) The nose butterfly mesh.

Figure 4 The current script truncates the trailing tip where the profile slope is 1. Triangles onthe truncated face are extruded downstream creating prism cells in the wake which minimizethe node count in this very small region.

Far Field Spacing Parameters

• Nominal distance to far field boundaries = 2ℓ: Baker [7] tried far field boundary distancesup to 5ℓ and found no changes to the overall forces beyond 2ℓ using the current boundaryconditions (see next section).

• Minimum distance through the unstructured transition region between the inner and outerstructured mesh = 2.0 hull radii.


• Far field circumferential dimension divisor = 2.0: adjusts far field circumferential dimensionrelative to that on the hull.

• Far field expansion factor r = 0.05.

Body Vortex Adaption

Figure 3a describes three parameters Φ, Rlee, Ylee that must be provided at several user suppliedaxial stations from vortex inception aft to the stern. The user positions these stations anddecides how many there will be. Since most spacing changes in the mesh take place gradually,these parameters do not need to be accurate. They can be supplied as tables but are easiest tospecify as empirical formulae. For the 4621 calculations, these parameters were set using:

Φ = 125.64 − 0.37908α + 0.3905αξ − 74.6385ξ4 (1)

Rlee = 1.00233 + 0.0039487α + 0.00177223α2ξ2 (2)

Ylee = 0.63675715 + 0.000481557(α − 8)2 − 0.5566942 (1 − ξ)4 (3)

for α in degrees and ξ = 0.3, 0.488, 0.722, 0.85, and 1.0. (ξ = 0 at the hull nose and 1.0 at thestern.)

Final Mesh Characteristics

Table 2 lists the characteristics of the meshes used in the 4621 study.

α (degrees) 0 6 12 18 18 18Mesh Scale 1.0 1.0 1.0 1.0 0.5 2.0

Nodes 2,849,900 3,336,246 3,989,537 4,664,647 2,335,004 9,286,159Elements 3,860,723 4,694,136 5,972,894 7,408,180 3,670,652 14,661,089

Hexahedra 2,562,834 2,974,220 3,492,835 4,007,066 1,999,692 8,044,686Tetrahedra 1,244,390 1,660,158 2,409,278 3,318,383 1,622,519 6,495,637Pyramids 22,747 26,154 30,047 34,284 21,540 54,564Prisms 30,752 33,604 40,734 48,447 26,901 66,202

Faces 126,683 135,696 151,209 165,377 104,379 262,530

Table 2 Model 4621 mesh entity counts for each simulation case.

Next Version Improvements

The meshing algorithm has successfully generated meshes for two hull shapes (STR and 4621)at up to 30 degrees incidence providing virtually trouble free calculations. Minor convergenceproblems occurred with two calculations, as discussed below and in the next section.

Improvements that could be made are as follows:

1) Adjust the nose spacing algorithm so spacing is chosen based on a minimum specifiedspacing (as is now done) and the spacing defined by the maximum turning angle of avector tangent to the surface. The 4621 model has a relatively sharp nose requiring finerspacing than does the STR nose.


2) Reduce the radial expansion through stages 1 and 3 from r = 0.15 to 0.1. The Spalart-Allmaras turbulence model calculation required special attention to complete its conver-gence because of, it is thought, excessive radial spacing at the tail near the surface. Therewere no problems with the other turbulence models.

3) The far field mesh needs some topological changes to allow the far field nodal densityto decrease faster than is currently possible. It is also thought that a much greater useof prisms in the transition layer, especially downstream of the hull, would unconstraindownstream axial spacing and reduce the high element count in the transition region.

3 RANS Simulation

The ANSYS CFX–10 steady-state RANS calculations used the following settings.

• The reference pressure for the calculation was set to 0 Pa.

• Buoyancy effects (static pressure depth gradients) were not modelled.

• The material fluid was liquid water with a dynamic viscosity of 8.899 × 10−4 kg/m/s anda density of 997.0 kg/m3.

• Double precision calculations were used throughout.

• Unless otherwise noted, the blending parameter was set to 1 so that pure second orderdiscretization was used.

• A timestep of 0.05 s was used for the first 10 iterations and 0.5 s for each iteration thereafter.

• Convergence was based on reducing the maximum of the normalized residuals of the mo-mentum and continuity (u, v,w, p) equations to less than 10−4.

The boundary conditions set on the far field mesh domains of Figure 2a are:

1) Inlet: normal flow at 2.283 m/s. The ‘low turbulence intensity’ option was used, which setsthe inlet turbulence intensity to 1% and the turbulent to molecular viscosity ratio to 1.

2) Symmetry plane: symmetry condition.

3) Outlet: Average static pressure over outlet set to 0 Pa.

4) Opening: top, bottom, and side use the CFX ‘static pressure for entrainment’ boundarycondition which allows flow in and out of the opening. A constant static pressure (0 Pa)is applied everywhere over the opening and the velocity gradient perpendicular to theboundary kept at zero. The turbulence gradient across the opening is also set to zero.This condition is most accurate when the flow is closely tangential to the boundary.

5) Wall: a smooth-wall, no-slip condition is applied on the hull surface.

Convergence was generally good, as shown in Figure 5. There were two exceptions. One wasthe Hi-Resolution RSM-BSL run where the maximum residual was left at 2 × 10−4 after 200iterations. The second was the Spalart-Allmaras turbulence model run where the maximumresidual stalled at 1.3×10−4 until the iteration timestep was reduced by two orders of magnitude.


0 30 60 90 120 150

Iterations

−5

−4

−3

−2

−1

0lo

g10(M

axiu

mN

orm

alize

dR

esid

ual)

u

v

w

p

0 30 60 90 120 150

Iterations

−5

−4

−3

−2

−1

0

log10(M

axiu

mN

orm

alize

dR

esid

ual)

Reynolds stresses:uu, uv, uw, vv, vw, ww

ω-turbulent frequency

Figure 5 Convergence history at α = 18 degrees, using the RSM-BSL turbulence model, puresecond order discretization, and mesh scaling = 1. Note that the convergence criteria areapplied only to the momentum and continuity equations on the left.

4 Overall Force and Moment Results

The Model 4621 experimental overall force and moment data were acquired from conventionaltowing tank tests using force gages, as described by Randwijck and Feldman [3]. The RANSforces and moments are automatically calculated on a ‘wall’ (the submarine half-hull in ourcase) by ANSYS CFX with the same accuracy as is used by the solver — second order for mostof our current calculations. This is important as it means the RANS force should converge ata second order rate in the verification study carried out below.

Tables 3 and 4 list the overall force and moment experimental results and compare themwith the RANS predictions. Table 3 does so for the SST and RSM calculations and Table 4 forthe alternate turbulence models. Figure 6 plots these data.

The error in the experimental data is not quoted. Estimates of the uncertainty in theZ and M measurements are obtained by calculating the standard deviation of the ±α valuesabout their means [8]:

σZ′ =

√

√

√

√

√

√

√

18∑

α=2α even

(

|Z ′| − |Z ′|)2

Nα

= 0.03 × 10−3 (4a)

σM ′ =

√

√

√

√

√

√

√

14∑

α=2α even

(

|M ′| − |M ′|)2

Nα

= 0.02 × 10−3 (4b)

where Nα = 9 for (4a) and 7 for (4b). These estimates will be low but are the best that can bedone with the information available.


Normal Force Z Pitching Moment M

αdeg

.

Sca

ling

discr

etiz

.ord

er Experiment

Z′ × 1000

SST

Exp.

RSM

Exp.

Experiment

M ′ × 1000

SST

Exp.

RSM

Exp.

6 1.0 2nd −1.10 0.77852 0.92386 1.245 1.05767 1.01565

12 1.0 2nd −3.45 0.74244 0.88427 2.18 1.09186 1.03377

18 1.0 2nd −7.30 0.77113 0.88405 2.88 1.09639 1.04875

18 0.5 2nd −7.30 0.77006 0.88536 2.88 1.09633 1.04689

18 2.0 2nd −7.30 0.77133 0.88372 2.88 1.09682 1.04976

18 1.0 Hi-Res −7.30 0.77941 0.89207 2.88 1.09327 1.04651

Table 3 Overall normal force and pitching moment experimental results and RANS predictions.The moment is about the center of buoyancy at ξ = 0.4456. Experimental values are averagesof values given by Sung [2] at ±α.

Turbulence Model

Normal Force

RANS Z′

−7.30 × 10−3

Pitching Moment

RANS M ′

2.88 × 10−3

Spalart-Allmaras 0.7224 1.1230

k-ǫ 0.5937 1.2032

RNG k-ǫ 0.5827 1.2067

Wilcox k-ω 0.7158 1.1238

BSL k-ω 0.7442 1.1126

SST 0.7711 1.0964

Table 4 Overall normal force and pitching moment RANS predictions relative to experimentfor alternate turbulence models, for α = 18 degrees, mesh scaling = 1, and second order dis-cretization.

Verification

Here we assess the accuracy of the numerical solution to the mathematical model (the RANSequations). The accuracy is determined by the discretization error in the mesh scaling = 1solutions. This is assessed using Richardson extrapolation on the three α = 18 degree solutions(mesh scaling = 0.5, 1.0, 2.0), as discussed by Roache [9] and endorsed by the Journal of FluidsEngineering [6].

The first step is to determine if the solutions asymptotically approach the true solution tothe mathematical model as the mesh is refined. If so, the observed order of convergence shouldbe consistent with the order of accuracy of the discretization method used (second order). The


0 3 6 9 12 15 18

α degrees

−8

−7

−6

−5

−4

−3

−2

−1

0

Z′×

1000

◦◦

◦

◦

◦

◦

◦

◦

◦

◦ ◦◦ ◦

◦ ◦

◦

◦

◦

◦

◦

◦

⊙

⊙

⊙

⋄

△▽

×+

◦ Experiment, R = 11.7 million

RANS Turbulence ModelRSM-BSL

⊙ SST

+ BSL k-ω

⋄ Spalart-Allmaras

× Wilcox k-ω

△ k-ǫ

▽ RNG k-ǫ

0 3 6 9 12 15 18

α degrees

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

M′×

1000

◦

◦

◦

◦

◦

◦◦

◦

◦

◦◦

◦

◦

◦

◦

◦

◦◦

◦

⊙

⊙

⊙⋄

△▽

×+

Figure 6 Overall force and moment for both experiment and RANS predictions as a func-tion of incidence angle and turbulence model. Mesh scaling = 1, second order discretization.Negative angle-of-attack experimental data are reflected into the positive angle-of-attack quad-rants. One standard deviation of experimental error is estimated at about 1/4 the height ofthe experimental data symbol (see text).


observed order of convergence is:

p =

ln

(

f3 − f2

f2 − f1

)

ln r(5)

where f3, f2, f1 are solutions from the coarse, medium, and fine meshes respectively, and r =h3/h2 = h2/h1 = 2 is the ratio of average mesh cell size. The results are shown in Table 5.

f SST RSM-BSL

Z 2.4 2.0

M −2.8 0.9

Table 5 Observed order of convergence p (from (5)) at α = 18 deg using scaling = 0.5, 1.0, 2.0.

Unfortunately, the accuracy of the Table 5 p values is questionable because the valuesare based on differences between numbers that are very close together. These numbers arelisted in Table 3 with five decimal places so the differences in (5) have reasonable resolution.However, the convergence criteria provides less than three decimal places of accuracy in thenumerical force predictions. For example, when the 18 degree, mesh scaling = 1, second orderdiscretization solution is converged until the maximum normalized residual is less than 10−5

instead of the usual 10−4, then the Z/SST entry of 0.77113 in Table 3 changes to 0.77045 andthe associated p value in Table 5 changes from 2.4 to −1.2. The normal force p values in Table 5are reasonably good probably because the solutions were converged to consistent, if inadequate,levels. Either this same technique has not worked for pitching moment or the solutions are not,in fact, in the asymptotic range. We cannot know for sure without converging the solutionsseveral more orders of magnitude.

If the solutions are in the asymptotic range, then Richardson extrapolation estimates theerror in the mesh scaling = 1 solution as:

∆f2 =f2 − f3

rp − 1(6)

Roache [9] defines a Grid Convergence Index (GCI) by multiplying this error by 3, a rough,empirically chosen number he admits is overly conservative but which is useful for historicalreasons. He suggests a ‘2σ’ error band (two standard deviations = 95% probability of beingcorrect) could be achieved using a factor between 1 and 3. The current preference is to workwith standard deviation and so (6) is used as an estimate for σ, one standard deviation.

Taking p = 2 in (6) results in the following estimates for the mesh discretization uncertainty(subscript d) at α = 18 degrees for the SST and RSM turbulence models:

σZ′d

∣

∣

∣

SST= 0.003 × 10−3 σZ′d

∣

∣

∣

RSM= 0.003 × 10−3

σM ′d

∣

∣

∣

SST= 0.00006 × 10−3 σM ′d

∣

∣

∣

RSM= 0.002 × 10−3

(7)

Again, it only makes sense to use these estimates if the solutions are converged enough thatthe fi − fi+1 difference in (6) has significant digits. The above mentioned Z/SST case which


was converged an extra order of magnitude indicates that the convergence error in Z ′ is at least0.005 × 10−3 and in M ′ is at least 0.0006 × 10−3. These quantities are larger than the SSTerrors in (7) and larger than the difference between the mesh scaling = 1.0 and 2.0 solutions. Apractical estimate of the verification uncertainties (subscript v) might simply use the maximumof the above noted quantities:

σZ′v = 0.005 × 10−3 σM ′v = 0.002 × 10−3 (8)

An important conclusion to be drawn from this section, then, is that the fine meshes usedin the current RANS calculations require an exceptionally high level of convergence to verifythe numerical solutions they give rise to. This level of convergence was not sought in this study.A fine mesh makes for a lot of work if verification is to be carried out properly.

Of course, the meshes were made fine to address visible discretization error in localizedquantities (Watt et al [4]) and to help ensure the solutions are in the asymptotic range. Thislatter point is a major concern. Baker [7] describes the process of developing and assessingthe current script and the meshes built with it for the DRDC STR hull. He obtained similarresults to those shown herein for α up to 30 degrees. However, Baker also built much largermeshes by refining them through localized regions of the flow (notably, smaller y+ spacings,stage 1 expansion rates, and axial spacings over the hull). He built meshes with as many as 16million nodes, the largest that can be generated with the 32-bit version of Gridgen. In a coupleof cases, he achieved increases in Z ′ magnitudes at α = 30 degrees (SST turbulence model) ofmore than 5% relative to the standard 4 to 5 million node mesh. This suggests the currentmeshes are not fine enough.

While verification has not been carried out as satisfactorily as desired, it seems likely thatdiscretization error is much less than the overall error in the present predictions. The SST Z ′

error in Table 3 is greater than 20% which is much larger than even the 5% increase in Z ′

magnitude suggested by Baker’s α = 30 degree STR results.

Validation

For predictions to be validated by experiment, the experimental error in the data and dis-cretization error in the predictions must be known. These are estimated and presented above.Since the experimental and discretization errors are much smaller than the differences shown inTables 3 and 4, our RANS predictions clearly cannot be validated. This means that modellingerror is present in the RANS equations at a level at least as large as the differences in the Tablesminus the experimental and discretization error.

The above error analysis also allows us to conclude that the apparent superiority of theRSM-BSL turbulence model is legitimate, since the difference between the RSM and SST pre-dictions are greater than the combined experimental and discretization error. Similarly, theSpalart-Allmaras one-equation model is better than the k-ǫ two-equation models and about asgood as the k-ω models, for this application at least. What is encouraging for ANSYS CFXusers is that the turbulence models recommended by the user documentation [5] are actuallythe best turbulence models to use for the current application.

A lack of validation does not mean the RANS predictions are not useful. It merely meansthat our analysis allows modelling error in the RANS equations to be resolved. Engineeringjudgment is required to decide whether use of the RANS model is acceptable in any situation.And in this regard, we note that the current RANS predictions provide among the best physicsbased predictions of these high Reynolds number separated flows that are available.


5 Normal Force DistributionsThe Model 4621 experimental data set is unique in that it provides normal force as an axialdistribution over the model. This was accomplished by cutting the hull into 10 sections andreassembling the sections with load cells between them [3]. The experimental measurementsand RANS predictions of the normal force on each section are compared in Figure 7.

Experimental error is not quoted for the distributed force measurements. It is estimatedby assuming the error in the sum of the 10 individual section forces (Z ′

summed=

∑10

i=1Z ′

i) ismuch greater than the error in the overall normal force obtained from a single measurement(as quoted in Table 3). The uncertainty in the summed section forces is therefore:

σZ′

summed

=

√

√

√

√

√

∑

α=6,12,18

(

Z ′summed − Z ′

Table 3

)2

3= 0.27 × 10−3 (9)

and this must equal the RMS sum of the 10 section measurement uncertainties:

σZ′

summed

=√

10σ2Z′

i

(10)

This gives the uncertainty in each section force measurement as:

σZ′

i

= 0.08 × 10−3 (11)

which is more than twice that of the overall normal force uncertainty (see (4a)). The uncertainty(11) is large enough that the differences between experiment and the RANS predictions inFigure 7 at α = 6 degrees are not meaningful. The differences over the afterbody at α = 12degrees for the SST turbulence model are meaningful as are all the differences on the afterbodyat α = 18 degrees. The large differences right at the nose section are meaningful at bothα = 12 and 18 degrees, assuming some systematic error is not present here. This assessmentassumes discretization error, which is difficult to quantify without further converging the RANSsolutions, is small, which may not be the case.

In viewing these distributions, it is worth keeping in mind that the overall normal forceon an isolated body in a potential (inviscid) flow is zero. The negative normal force over theforebody is canceled by an equal and opposite positive force on the afterbody. A relativelylarge pitching moment results. In a real flow, viscosity results in separation from the leewardafterbody preventing the pressure recovery experienced in an ideal flow. This reduces thepositive normal force over the afterbody, resulting in a negative overall normal force and areduced pitching moment.

The Figure 7 comparisons show that the negative forebody forces are relatively well pre-dicted by the RANS methods, with the possible exception of the nose itself. However, theafterbody forces are overpredicted, at least at moderate to high incidence angles where Fig-ure 6 shows large overall force and moment discrepancies occur. This suggests that RANSpressure recovery predictions are excessive and, therefore, that energy loss in the wakes is beingunderpredicted. This is consistent with total pressure comparisons presented by Watt et al [4]

showing substantially more energy loss measured in the cores of the body vortices in experi-ments with the DRDC STR hull than are predicted by the RANS SST method. Table 6 showshow total pressure minima decrease in the vortices in the current RANS predictions as theturbulence model improves.


0−4

−3

−2

−1

0

1

Z′×

1000

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

ξ ξ

0.0 0.2 0.4 0.6 0.8 1.0

ξ

◦ Experiment, R = 11.7 million

RANS Turbulence ModelRSM-BSL

⊙ SST

◦

◦

◦◦

◦

◦ ◦◦

◦◦

⊙

⊙

⊙

⊙

⊙

⊙⊙ ⊙

⊙

⊙

α = 6

◦

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◦

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◦◦

⊙

⊙

⊙

⊙

⊙

⊙

⊙⊙

⊙

⊙

α = 12

◦

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◦◦

◦

⊙

⊙

⊙

⊙

⊙

⊙

⊙ ⊙⊙

⊙

α = 18

//

Figure 7 Experimental measurements and RANS predictions of the normal force distributionsover the length of the hull at α = 6, 12, and 18 degrees. Mesh scaling = 1, second orderdiscretization. The uncertainty in the experimental data is estimated to be plus or minus theheight of the symbol used for this data.

Model 4621 RANS PredictionsMinimum total pressurein the vortex core

at ξ = 1Spalart-

AllmarasSST RSM-BSL

p − p∞ + 1

2ρU2

1

2ρU2

∞

0.645 0.596 0.530

Table 6 RANS predictions of the energy in the core of the body vortex at the stern.


6 Concluding Remarks

Modelling error exists in the ANSYS CFX RANS predictions of the flow about the Series 58Model 4621 unappended submarine hull. Much of this error is associated with the turbulencemodelling. The Shear Stress Transport turbulence model is the best two-equation model butstill underpredicts normal force by more than 20%. The Baseline Reynolds Stress model doesmuch better with an underprediction just over 10%. This improvement is likely due to theability of the RSM model to predict anisotropic turbulence, the cost of which is solving seventurbulence model equations instead of two.

This conclusion was made possible by an extensive effort to minimize and quantify dis-cretization error asociated with the mesh. This effort was compromised somewhat by notconverging the RANS calculations thoroughly enough —doing so would likely require reducingthe maximum normalized residuals another two orders of magnitude, to less than 10−6. Nev-ertheless, the current verification effort was successful enough to allow modelling error to beidentified and it has shown what needs to be done to carry out a proper verification analysis inthe future.


References

1. C.R. Baker, T.L. Jeans, A.G. Gerber, A.G.L. Holloway, G.D. Watt, “Examination of theFlow Characteristics Around a Streamlined Axisymmetric Shape,” Proceedings ofFEDSM2005 ASME, Houston, June 2005 (FEDSM2005–77149).

2. C.-H. Sung, “Validation of Forces, Moments and Stability Derivatives of a ManeuveringSeries-58 Bare Hull,” 25th Symposium on Naval Hydrodynamics, St. John’s, August 2004.

3. E.F. Van Randwijck and J.P. Feldman, “Results of Experiments with a Segmented Modelto Investigate The Distribution of the Hydrodynamic Forces and Moments on a StreamlinedBody of Revolution at an Angle of Attack or with a Pitching Angular Velocity,” NSWCCD–50–TR–2000/008, February 2000.

4. G.D. Watt, C.R. Baker, A.G. Gerber, C. Fouts, “Scripted Hybrid Mesh Adaption for HighIncidence RANS Flows About Axisymmetric Shapes,” 44th AIAA Aerospace Sciences Meet-ing and Exhibit, Reno, 9–12 January 2006.

5. ANSYS Canada Ltd., ANSYS CFX–10, Service Pack 1 user documentation, 2006.

6. Journal of Fluids Engineering Editorial Policy Statement on the Control of Numerical Accu-racy, http://journaltool.asme.org/Content/AuthorResources.cfm, accessible as of May 2006.

7. C.R. Baker, “A Strategic Meshing Approach to Modelling Hydrodynamic Flow AroundStreamlined Axisymmetric Shapes,” Masters Thesis, Department of Mechanical Engineering,University of New Brunswick, March 2006.

8. J.R. Taylor, An Introduction to Error Analysis, 2nd edition, University Science Books, 1997.

9. P.J. Roache, “Verification and Validation in Computational Science and Engineering,” Her-mosa Publishers, 1998.



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1. ORIGINATOR (the name and address of the organization preparing the document.Organizations for whom the document was prepared, e.g. Centre sponsoring acontractor's report, or tasking agency, are entered in section 8.)

DRDC Atlantic

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3. TITLE (the complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S,C,R or U) in parentheses after the title).

ANSYS CFX–10 RANS Normal Force Predictions for the Series 58 Model 4621Unappended Axisymmetric Submarine Hull in Translation

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Watt, George D., Baker, Christopher R., Gerber, Andrew G.

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The ability of the commercial Reynolds Averaged Navier Stokes (RANS) software packageANSYS CFX-10 to predict the normal force on an unappended submarine hull at uniformincidence angles up to 18 degrees is evaluated. Hybrid meshes of several million nodeseach are created using a script to adapt the mesh to the body vortex for various angles ofattack. A formal verification procedure was attempted and shown to be dependent onsolutions converged to four or five significant figures. Numerical discretization error is shownto be much smaller than the differences between experimental and RANS normal forces.Turbulence modelling has a marked effect on these differences, which are greater than 20%for the Shear Stress Transport two-equation model and 10% for the Baseline ReynoldsStress seven-equation model.

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Submarine hydrodynamicsReynolds Averaged Navier-Stokes equationsTurbulence modellingVerification and validation


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ANSYS CFX–10 RANS Normal Force Predictions for the ...ANSYS CFX–10 RANS Normal Force Predictions for the Series 58 Model 4621 Unappended Axisymmetric Submarine Hull in Translation