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High-Incidence and Dynamic Pitch-Up Maneuvering Characteristics of a Prolate Spheroid - CFD Validation S. -E. Kim, S. H. Rhee, (Fluent Inc., Lebanon, NH 03766, U.S.A.) and D. Cokljat (Fluent Europe Ltd., Sheffield, U.K.) ABSTRACT This paper presents the results of a computational study of the flow around a 6:1 prolate spheroid at a range of incidence angles and during a dynamic pitch- up maneuvering. Several engineering turbulence mod- els in popular use today are employed for turbulence closure. Attempts are made to improve the perfor- mance of second-moment closure models with modi- fied length-scale equations. The computational results are compared with the experimental data in terms of crossbow separation pattern, pressure, skin-friction and wall-shear angles on the body surface, and lift and pitch- ing moment characteristics. The prediction accuracy varies widely depending on the turbulence model em- ployed. Considering the challenging nature of the flow, the fidelity of the predictions shown by some turbulence models such as Wilcox' k-ce model and the Reynolds- stress transport models with modified length-scale equa- tions are highly commendable. INTRODUCTION Despite its simple geometry, flow around a pro- late spheroid in maneuvering carries a rich gallery ex- hibiting a variety of complex three-dimensional tur- bulent shear flows, featuring stagnation flow, highly three-dimensional boundary layer under the influence of strong pressure gradients and streamline curvature, cross-flow separation, and formation of free-vortex sheet and ensuing stream-wise vortices. All these fea- tures of spheroid flows are the archetypes of flows around airborne and underwater bodies at incidence or in maneuvering, warranting an in-depth study. Flow past prolate spheroids has been studied by many others experimentally and numerically. Among the most relevant to the present study are the early works of Meter et al. (1984, 1986) and the more recent works of the group at Virginia Polytechnic Institute (VPI) (Chesnakas and Simpson, 1997; Wetzel et al., 1998; Goody et al., 2000~. The series of experimental studies conducted by these two groups provide most compre- 1 Fig. 1 Cross-flow separation and streamwise vortices on a 6: 1 prolate spheroid at a = 30°: The pathlines are computed using the present CFD solution. hensive experimental data, revealing important physics of the flow and offering an invaluable data-set useful to validate computational fluid dynamics (CFD) codes. On the numerical side, there have been quite a few studies using reduced Navier-Stokes equations in ear- lier days and, more recently, using full Naiver-Stokes or Reynolds-averaged Navier-Stokes (RANS) equations (Vatsa et al., 1989; Deng et al., 1990; Kim and Patel, 1991; Gee et al., 1992; Zheng et al., 1997; Rhee and Hino, 2000~. Yet most of these numerical studies are not quite up-to-date and comprehensive enough to give a clear perspective as to what can be achieved by today's CFD. The present paper is geared toward contributing to the ship hydrodynamics community a comprehensive sta- tus report on the subject flow which will enable one to gage what today's CFD has to offer in predicting the turbulent shear flow around a 6:1 prolate spheroid. To that end, we chose the case measured by the group at VPI. The present study covers the steady flow for the entire range of incidence angle experimentally investi- gated (a = 10° ~ 30°), and the unsteady flow associated
with the dynamic pitch-up maneuvering as well. As with other complex three-dimensional turbulent shear flows, turbulence modeling plays a significant role, af- fecting the prediction accuracy, especially in light of the challenging features of the subject flow noted ear- lier. In pursuit of the best possible predictions, sev- eral most popular engineering turbulence models with good track-records for similar flows were employed, in- cluding an eddy-viscosity transport model (Spalart and Allmaras, 1994), two different versions of k-ce models (Wilcox, 1998; Menter, 1994), and second-moment clo- sure (SMC) models. In addition to assessing the fidelity of these models for the present flow, we will also ex- plore some avenues to improving the performance of the selected models. All the computations were carried out using a finite-volume discretization based a RANS solver (Mathur and Murthy, 1997; Kim et al., 1998~. The paper is organized as follows. We start by look- ing at some of the salient features of the flow and pon- dering upon their implications to turbulence modeling of the subject flow. The numerical results are utilized to illustrate some of the significant characteristics of the flow. This is followed by a brief overview of the numerical method and turbulence models employed in this study. Finally the computational results will be pre- sented along with the discussions. PREVIEW OF MAIN FLOW FEATURES How can we characterize the present flow? A vi- sual impression of the mean flow in question is aptly portrayed by Figure 1, which was generated using the numerical solution for the a = 30° case. The figure serves nicely to highlight the most prominent features of the flow, i.e. crossbow in the boundary layer, free vortex sheet, and stream-wise vortices. The whole phenom- ena depicted here are an embodiment of the so-called "crossflow" or"open" separation, the significance of which can be recognized from the fact that the struc- ture of separation and its change with incidence angle greatly affect maneuvering characteristics of the body such as forces and moments acting on it. Another mo- tivation to characterize the mean flow is that a good understanding of it, if qualitative, often enables one to surmise whether or not certain turbulence models will be adequate for the flow at hand. In this vein, one of the useful questions to ask for the present flow is: how the mean flow deforms? Much insight to the mean flow along this line can be gained from contours of a normalized invariant of de- formation tensor (Hunt, 1992) defined as: SijSij - QijQij SijSij + QijQii 2.95e~01 1.19~01 -5.76e-0. -2.34e~1 -4.1 Oe-O] -5.86e-01 -7.63e-01 1.OOe+O _ ~ _ .24c01~_ Prolate Spheroid at alpha = 20 deg. Contours of deformation-ratio Section at x/L = 0.772 Fig. 2 Contours of the deformation invariant (see Equa- tion (1) for its definition) in the crossflow plane at x/L = 0.772 and a = 20° computed using the RSTM-2 result where Si; and Qij are strain and rotation tensors defined by Sij(aui/axj+aui/Oxi)/2 and Qij (OUi/Oxj - OUj/Oxi) /2, respectively. This invariant, which ranges between -1 and 1, is a convenient measure of the relative importance of strain, shear, and rotation. Note that ~ = 1 for pure strain, Z) = 0 for pure shear, and Z) = - 1 for pure rotation. Figure 2 shows the contours of this invariant in the crossflow plane at x/l = 0.772 and the incidence angle of a = 20°. According to the figure, the subject flow is largely shear-dominated (~) ~ O) in the leeward bound- ary layer. However, the flow becomes predominantly rotational (1) ~1) near the core of the stream-wise vortices. Evidently, the mean flow portrayed here is highly three-dimensional and carries significant extra rates of strain and/or rotation, providing an acid test for turbulence models. As regards the effects of rotation on turbulence at the most fundamental level, say, ho- mogeneous isotropic turbulence, it is a well-established fact that rotation inhibits energy transfer from larger to smaller eddies, decreasing the decay rate of TKE (Wige- land and Nagib, 1978~. It is also well known that, when a mean shear exists, rotation can either delay or acceler- ate the energy transfer depending on the relative orienta- tions of the mean shear and the rotation, attenuating or accentuating turbulence accordingly. The well-known effects of streamline curvature, either convex or con- cave, which is also relevant to the subject flow, can be explained in the same way. How rapid the mean flow is strained, sheared or ro- tatin~ is also of interest from a turbulence modeling standpoint. Time-scale of mean flow (1/S or 1/Q) normalized by turbulence time-scale (k/£), i. e., Sk/£ (S_ ~/~) and Qk/£ (Q_ ~/~), are good measures for that. The contours of these "relative" strain (1) and rotation in the crossflow plane at x/L = 0.772 are 2
3.24e+01 2.91e+01 2.59e+01 2.27e+01 1.94e+01 1.62e+01 1.30e+01 9.72e+0C 6.48e+0C 3.24e+0C 3.96e-04 6:1 Prolate Spheroid at alpha = 20 deg. Contours of relativ~strain AIL = 0.772 Fig. 3 Contours of relative strain (Sk/) at x/L = 0.772 (a = 20°) computed using the RSTM-2 result 1 .23e+01 1.1 1e+01 9.83e+0C 8.60e+0C ~~ .~. 7.37e+0t 6.1 4e+0C 4.91 e+OC 3.68e+OC 2.46e+00 1.23e+00 2.01 e-04 6:1 Prolate Spheroid at alpha = 20 deg. Contours of relative-rotation x/L = 0.772 Fig. 4 Contours of relative rotation (Qk/~) at x/L = 0.772 (a = 20°) computed using the RSTM-2 result shown in Figure 3 and Figure 4. Notwithstanding the lack of absolute accuracy of these quantities, the level of these two quantities (Sk/S, Qk/e >> 1) as shown in the figures suffices to indicate that the subject flow is dis- torted quite rapidly. This has negative implications for almost all aspects of turbulence modeling. Both eddy- viscosity model (EVM) and SMC suffer when the mean flow under consideration is rapidly sheared, strained, or rotating (Speziale, 1999~. All these considerations on the mean flow set the stage for the following question: how far the turbulence in a given flow is from equilibrium? It is an important question to ask, inasmuch as almost all the phenomeno- logical turbulence models in use today, including most sophisticated SMC-based models, rely upon an equilib- rium assumption in one way or another. Equilibrium refers to a state where production of turbulent kinetic energy (TKE), which is an event involving larger ed- dies, is balanced by its dissipation rate occurring at small scales. In highly complex turbulent flows like the present one, however, an equilibrium state is less than likely to be attained. Several mechanisms are respon- sible for causing nonequilibrium of turbulence. In the present flow, the salient features of the present flow dis- cussed so far act to make the turbulence to depart from equilibrium. It is surmised that rapid strain and rotation, streamline curvature, and adverse pressure gradient in the boundary layer play significant roles. Thus, tur- bulence models that are better capable of representing these physics and their effects on turbulence are likely to be more successful for the present flow than others which do not. The experimental studies cited in the beginning ex- tensively discuss and provide valuable insight into the salient features of the present flow. Among many oth- ers, we find particularly noteworthy the discussion by Chesnakas and Simpson (1997) regarding turbulence anisotropy based on their measurements of the individ- ual Reynolds stresses and the mean velocity field. By analysing the stresses and mean rates-of-strain, they at- tempted to assess the validity of isotropic eddy-viscosity assumption for the present flow. They found that the Reynolds stresses are largely aligned with the strain rates inside the boundary layer at a low incidence angle (a = 10°~. However, they become grossly misaligned almost everywhere else, especially along the free vortex sheet and near the vortices on the leeward side of the body and at high incidence angle. As they concluded, this suggests that turbulence models based on isotropic eddy-viscosity are likely to perform poorly, warranting use of SMC. All these observations and thoughts influenced our overall turbulence modeling strategy adopted in this study; the model selection and the rationales behind our attempts to modify some turbulence models. MATHEMATICAL MODELING Governing Equations - Ensemble-Averaged Navier-Stokes (EANS) equations The present study adopts ensemble-averaged Navier- Stokes equations as the governing equations, since the salient features of the flow we are mainly interested in can be found in averaged or mean flow. It should be em- phasized that the ensemble averaging retains the local time-derivative of the averaged velocity and other scalar fields. Use of unsteady ensemble-averaged Navier- Stokes equations, therefore, enables one to tackle flows where there are significant temporal variations of aver- aged flow fields such as alternate vortex-shedding be- hind bluff-bodies that can occur in the present flow at high incidence. In spite of its relevance to the present flow at high incidence, vortex shedding will not be ad- dressed in this paper. Yet the unsteady EANS equations
are needed to compute the unsteady flow due to the dy- namic pitch-up motion. To simulate the dynamic pitch-up motion, we solved the governing equations in a non-inertial coordinate sys- tem which rotates with the body at the prescribed angu- lar velocity. Use of non-inertial frame greatly simplifies the application of boundary conditions. However, the body-force terms arising from the use of non-inertial frame of reference should be accounted for. For the dynamic pitch-up maneuver considered here, the body- force terms are due to Coriolis, centripetal and angular accelerations and can be written as: fb = - 2Q x VQ x (Q x r) i, x r (2) where r and V are the the displacement and the fluid velocity vectors in the non-inertial (rotating) coordinate system, and Q is the angular velocity of the non-inertial coordinate system. TURBULENCE MODELS Among many choices, we screened the turbulence models based on their relevance and track-records for aerodynamics applications. So the k-£ models are omit- ted in the paper. Table 1 summarizes the models chosen for the paper. The list includes four EVM and three SMC models. The details of the models can be found in the cited papers. Abbreviation Description SA Eddy-viscosity transport model (Spalart and Allmaras, 1994) SST Blended k-m Ik-£ (Menter, 1994) KO-1 High-Re k-m (Wilcox, 1998) KO-2 Low-Re k-o (Wilcox, 1998) RSTM-1 Second-moment closure (Gibson and Launder, 1978) Second-moment closure Shih et al.'s (1995) £ equation Second-moment closure with Durbin's (1990) £ equation RSTM-2 RSTM-3 Table 1 Turbulence models used in this study Eddy-Viscosity Transport Model of Spalart and Allmaras In the Spalart-Allmaras (SA) model, one directly solves the transport equation for an effective viscos- ity, v (Spalart and Allmaras, 19941. The SA model has become rapidly popular especially in the aerospace community due to its commendable performance for boundary layer flows subjected to adverse pressure gra- dient. The SA model used in this study is identical to the original model except one thing. In the SA model. the production of v is computed by: GV=PCb~SV where Cal is a model constant, and v the effective cosity. In the original model, S in Equation (3) is com- puted from a modulus of rotation rate tensor as: (3) vis- .. S= x/2QijQij _ Q (4) For thin boundary layer flows, it would be of no con- sequence whether the modulus of rotation-rate tensor or that of strain-rate tensor is used. However, adopting the vorticity magnitude has a potential to cause problems in swirling or vortical flows like the present flow where ro- tation dominates over strain in the vicinity of the cores of the streamwise vortices, as shown earlier (Figure 2~. The original S-A model indeed performed very poorly for the present flow. To avoid this, we adopted what Dacles-Mariani et al. proposed, i. e., 3 = Q + Cv min (O. SQ) (5) where S(_ >/~35) and Q(~) are the moduli of strain-rate and rotation-rate tensors, respec- tively, with Cv being an adjustable constant of an order of 1. We simply took the value Cv = 2.0 from the cited reference. k-m Models We adopted here three variants of k-ce models in pop- ular use today. The first two k-m models, denoted as KO-1 and KO-2 in Table 1, are based on the recently revised version (Wilcox, 1998~. The difference between the original model (Wilcox, 1988) and the revised one lies in the "shear-correction" and "vortex-stretching" terms which were added in the new model to improve the model performance mainly for free shear flows such as far-wakes, mixing layers, and jets. The new model also has a re-calibrated low-Reynolds number model de- signed to account for transitional effects. KO-1 and KO- 2 refer to the revised Wilcox' models without and with the low-Reynolds number modification, respectively. Another variant of k-m model adopted in this study is Menter's k-m model (Menter, 1994) often referred to as shear-stress transport (SST) k-m model in the litera- ture. The SST model is essentially a"two-zone" model that blends a variant of k-m model in the inner layer and what is tantamount to a traditional k-£ model in the outer layer. In addition, the SST model clips turbulent viscosity based on the argument that the structural sim- ilarity between k and ~uv~ (k/~al = 0.3) should be 4
preserved, which can be used to set an upper limit on turbulent viscosity as: vt = min ( k, at k) (6) It should be noted that, in the SST model, this limit is applied within the boundary layer only. As for the present flow, the leeward side of the flow will be mostly unaffected by the clipping, except the near-wall re- gion. And the model essentially reduces to a traditional k-£ model there. Second-moment closure models The baseline model of the Reynolds-stress transport models (RSTM) used in this study is largely based on the Rotta's model (Rotta, 1951) for the slow redistribu- tion term, isotropization of production (IP) model of Fu et al. (1987), and the wall-reflection model of Gibson and Launder (1978~. The baseline model, which will be called RSTM-1 hereafter, was implemented in an un- structured mesh based finite-volume RANS solver, and has been been validated for a number of complex three- dimensional internal and external flows (Kim, 2001; Kim, 2002~. The unique features of the implementa- tion include: an isotropic turbulent diffusion models for Reynolds-stress and dissipation equations, a high- order dissipation term designed to prevent decoupling of Reynolds stresses, and mean velocity field arising from co-located, cell-centered finite volume discretiza- tion scheme. The wall-reflection effects in the pressure- strain correlation were included with the aid of a wall- proximity function that allows wall distance and wall normals to be computed for arbitrary wall configura- tions. In the RSTM-1 model, turbulence length-scale is ob- tained by solving the transport equation for £ given by: D£ a + _ + c£ PiiC£2 Dr axj [( ~£) axj] 2 k P k (7) where Pij is the production of Reynolds stresses, ~£ = 1.3, Cal = 1.44, C£2 = 1.83. Despite more-than-modest improvements over linear k-£ models, the performance of the RSTM-1 model fell short of our expectation based on the remarkable perfor- mance of the same model for a similar flow (Kim, 2001~. Its rather disappointing performance begs a question as to what the main culprits could be. Among many leads alluded to earlier, including the shortcomings of the SMC itself for non-equilibrium flows like the subject flow, we decided to track down the £-equation. As dis- cussed earlier, the boundary layer and free vortex sheet with large shear/strain and strong rotation make con- ventional length-scale equations, i. e., the £-equation in Equation (7), less than adequate for the present flow. To investigate the impact of the length-scale equation on the prediction, we adopted here two alternative £ model equations. One of them is the £-equation proposed by the turbulence modeling group at NASA Glenn (Shih et al., 1995), which was developed starting from an exact equation for mean-square vorticity fluctuation (i). This enstrophy-based £ equation has been used mostly in the context of two-equation k-£ models, except for the study by Luo and Lashiminarayana (1997) who adopted it in conjunction with a RSTM to study duct flows. They reported that this new £ model equation was capable of predicting the turbulence enhancement observed in the boundary layer near concave walls. The new £ model equation was also claimed to better describe the process of vortex stretching and spectral energy transfer, and has actually been found to perform significantly better than traditional k-£ models for boundary layers involv- ing rapid strain and severe adverse pressure gradient. All these benefits seem relevant to the present flow. The new £-equation reads: PDt aXi [(lo 6e) axj] P ~ P 2k+ (8) where 5 - >/~), and Cat = max [0.43, '~ + 5], 11 = Sk/£ C2 = 1.9, C}k = 1.0, 6£ = 1.2 In the RSTM-2 model, the standard £-equation in the baseline model was replaced by the new £-equation in Equation (8~. The strong non-equilibrium turbulence in the present flow prompted us to look for models known to better deal with that. And we employed the modification pro- posed by Durbin (1990) and Chen and Kim (1987) as the second alternative. The modified £-equation is given by: Ds a + · _ + c£~PkC£2 t9' Dt axj [( ~£) axe] k P k where C,£ = 1.3, C£2 = 1.83, and Call is computed from: Cal = C£i (1 +aPk/£) (10) where Cal = 1.4 and a = 0.05. As can be noted, the model parameter, Call in the "production-of-dissipation" term is a function of Pk/~. This term adds more dissipa- tion as production of TKE becomes larger than dissipa- tion rate, suppressing spuriously large TKE frequently encountered in complex flows. The baseline model with the standard £-equation replaced by Equation (9) and Equation (10) is denoted as RSTM-3 in this paper. s
Near-wall treatment The choice of near-wall treatment depends on the res- olution of the near-wall mesh in use. For the present study, we employed both fine and coarse meshes. The fine meshes are such that the entire near-wall region is resolved down to wall including viscous sublayer. SA, SST, KO-1, and KO-2 models, all of which can be nat- urally integrated to wall, were run on the fine meshes. When fine meshes are used, the wall boundary con- ditions for the mean velocity and turbulent quantities essentially exploit no-slip condition at walls. For ce, we "fix" the asymptotic value of ce as y ~ O at wall-adjacent cells, using: 6v pyp2 where yp is the distance from the wall to the cell center, andp=0.075. The coarse meshes were designed to skip the viscosity-affected region and to place the first grid (cell- center) points in fully turbulent region. We then can employ wall functions, namely, the law-of-the-wall and related hypotheses, to derive the wall boundary condi- tions for the mean velocity and turbulence quantities (Kim, 1998; Kim, 2001) such as Reynolds stresses and co. All RSTM computations were made on the coarse mesh only. Among the EVMs, SST and KO-1 models were run on the coarse mesh using the wall functions. NUMERICAL METHOD A cell-centered finite-volume method is employed along with a linear reconstruction scheme that allows use of computational elements (cells) with arbitrary polyhedral topology, including quadrilateral, hexahe- dral, triangular, tetrahedral, pyramidal, prismatic, and hybrid meshes. The velocity-pressure coupling and overall solution procedure are based on SIMPLE-type segregated algorithm adapted to unstructured mesh. The convection terms are discretized using a third-order upwind scheme, and the diffusion terms using cen- tral differencing scheme. The high-order terms are treated using a deferred correction approach. The dis- cretized algebraic equations are solved using a point- wise Gauss-Seidel iterative algorithm. An algebraic multigrid method is employed to accelerate solution convergence. The details of the numerical method are described by Mathur and Murthy (1997), Kim et al. (1998), and Kim (2001~. COMPUTATIONAL DETAILS Body geometry and computational conditions The experiments conducted at the Virginia Poly- technic Institute with 1.37m-long model of a 6:1 pro- late spheroid are simulated in the computations. The Reynolds number (Re~) based on the freestream veloc- ity (Uo) and the body-length (L) is 4.2 x 106. In the ex- periments, the body was supported by a sting mounted at the rear-end of the body. The body was also mounted a trip at x/L = 0.2 to help trigger laminar-to-turbulent transition. The sting-mount were not modeled in the computations. However, we attempted to mimic the ef- fects of the trip using a numerical trigger. More on this will be discussed shortly in the beginning of the "Re- sults" section. The coordinate system adopted in this study is such that the positive x-axis is in the streamwise direction, `11y y points to the upward vertical direction, and the xy plane makes the vertical symmmetry plane. The origin of the coordinate system is placed at the fore-end of the spheroid. Solution domain, mesh, and boundary conditions The solution domain covers2.75 < x/L > 3.425 and2.75 < y/L, z/L > 2.75, in the streamwise and lateral direction, respectively. Single-block hexahedral meshes are used. Four different sizes of mesh were used to see the effects of different near-wall modeling strategies (wall function vis-a-vis near-wall-resolving approach) and also to ensure mesh-independency of the numerical solutions. Two of them (540,000 cells, 1,380,200 cells) were designed to penetrate the viscous- sublayer. Two other meshes (345,000 cells, 1,120,000 cells) were made for the wall function calculations. For each pair of meshes, mesh-dependency of the numerical solutions were found to be insignificant. Therefore, we present here the results for the respective coarser meshes : 345,000-cell mesh for wall function calculations, and 540,000-cell mesh for the near-wall model calculations. The y+ values at the wall-adjacent cells of the fine and the coarse meshes were near y+ = 1.0 and y+ = 30, re- spectively, over most of the body surface. The symmetry of the geometry and the flow allowed us to model only a half of the domain. Thus, the domain boundary consists of the body surface, upstream/far- field inlet, vertical plane of symmetry, and exit bound- ary. The wall boundary conditions were applied as de- scribed earlier. On the inlet boundary, freestream condi- tions were specified. On the exit boundary, the solutions variables were extrapolated. The steady computations were carried out for three incidence angles, namely, a = 10°,20O,30°. The nu- merical solutions were deemed converged when scaled residuals for all solution variables drop by four orders of magnitude. The lift and pitching moment were also monitored to ensure full convergence of the solutions. 6
RESULTS Effects of boundary layer tripping To mimic the effects of the physical trip mounted at x/L = 0.2, we employed a numerical trigger by which the flow in the upstream of the trigger was treated as being laminar, whereas the turbulence models were en- forced to become fully effective right at the trigger. Several runs were made with the k-co models with the numerical trigger. However. the numerical results did not show any meaningful differences from the fully- turbulent results in terms of the global features of the flow such as the surface quantities and the lift and mo- ment, except a very small region in the laminar region. Thus, we made all the subsequent computations assum- ing that the flow is fully turbulent over the entire body, which are presented in the paper. Crossflow separation pattern Line of crossbow separation is usually defined as a (hypothetical) line in a given wall-shear vector field onto which neighboring wall limiting streamlines con- verge. The separation line can be visualized by sur- face flow visualizations, as long as convergence of the wall limiting streamlines is strong enough. The VPI group looked at several other indicators (Chesnakas and Simpson, 1997; Wetzel et al., 1998) and studied the correlations between those and the separation location determined from the flow visualization. One of the proposed indicators was the location of minimum wall- shear (or skin-friction). The locations of skin-friction minima were found near the separation yet consistently on the further leeward side of the separation line. For comparison with the experimental data available at x/L = 0.729, two circumferential angles were read off from the numerical results. One is the angle at which the skin-friction becomes a minimum, which enabled us to make direct comparison with the measurement. Fig- ure 5(a) shows the results. It should be mentioned that all the EVM results shown here were obtained on the fine mesh, whereas the RSTM results are based on the coarse mesh. Quite a large scatter is observed among the results, with SA, KO-2 and RSTM-2 deviating far- thest from the experimental data. In the experiment, the location where the circumferential velocity changes its sign (direction) was found closer to the actual separation location. Figure 5(b) shows the angles thus-determined from the numerical results, along with the separation lo- cation determined from the oil flow visualization. In comparison to the minimum Cf angles in Figure 5(a), the angles determined this way shift leeward by a sub- stantial amount. The overall trends shown by the differ- ent turbulence models remain largely unchanged from what were seen earlier. Particularly noteworthy is that KO-2 and RSTM-2 models better predict the actual sep- aration locations than the locations of minimum Cf. It is also seen that the SA and SST models continue to predict a delayed separation. The experiments also reveal that, at higher incidence angle (e.g., a = 20°), another separation line emerges. The numerical results from some of the turbulence mod- els also showed a clear evidence for the secondary separation. Figure 6 depicts the limiting wall stream- lines numerically visualized using the result of KO-2 model. Note that the body in the figure was rotated around its axis to provide a better view of the surface flow pattern. Clearly visible toward the leeward sym- metry plane is a line starting from a little downstream of x/L = 0.6, onto which the neighboring streamlines converge. By x/L = 0.772, the secondary separation line becomes full-fledged, as Chesnakas and Simpson (1997) described in their paper. In addition, the diverg- ing streamlines between the two separation lines imply that the flow re-attaches to the body surface along a line (denoted as reattachment line in the figure). The reat- tachment line is located much closer to the secondary separation line. All these features predicted by the KO-2 and RSTM models are in good accordance with the experimental observations (Chesnakas and Simpson, 1997~. Surface quantities The behaviors of the surface quantities like pres- sure (Cp _ (ppo)/0.5pU02~' skin friction (Cf _ ~w/0.5pU02), and wall-shear angle flewarctan Wu ~ carry rich information not only on the surface flow itself but off-the-wall structures like the free vortex layer and streamwise vortices. At a = 10°, the lowest incidence angle we computed in this study, the crossbow and resulting separation are rather weak. And the differ- ences among the turbulence models were found to be also small, although the differences become more dis- cernible as one goes to the rear and leeward side of the body. Although not shown here, SA model grossly under-predicted the circumferential variations of all sur- face quantities, while the KO-2 model tends to overpre- dict them. Overall, the results of the SST and KO-1 models were about right. The behaviors of the predicted surface quantities appeared to be largely consistent with the experimental observations with regard to the incipi- ent crossbow at x/L = 0.6 and the fully-developed sep- aration line at x/L = 0.772. The surface quantities exhibit much richer features at or = 20°. And the differences among the models also become far more noticeable. Figure 7, Figure 8, and Figure 9 depict the circumferential distributions of Cp, Cf. and Do at a = 20°, respectively. It should be noted that the results shown in these figures were obtained 7
(a) Circumferential angle of minimum Cf 180 leeward be 150 120 . . Measured (men. Cf) 0 SA ° SST KO- ~ KO-2 · RSTM-1 (wall fn.) o RSTM-2 (wall fn.) · RSTM-3 (wall fn.) 7V5 10 15 20 25 30 35 Angle of attack (deg.) (b) Circumferential angle of circumferential vet. sign change 180 1 L L. I ~ . - . ~ · Oil flow t ~ r ~ SST . . . · KO-1 . . . '` Key ~ ~ , ~ ,, r ~ RSTM-1 (wall fn.) ~ To i 0 RSTM-2 (wall fn.) -:~ i-: :~ :::::::-::::: · RSTM-3 (wall fn.) . . . m I I 120 ~ i Y 90 . ~ t i 5 10 15 20 25 Angle of attack (deg.) O -0.1 30 35 Fig. 5 Change in circumferential location of crossflow 0 3 separation with incidence at x/L = 0.729 determined by (a) minimum Cf (b) circumferential velocity using the fine (near-wall-resolving) mesh. The exper- imental data (Chesnakas and Simpson, 1997; Wetzel et al., 1998) show that the primary vortex is located near ~ = 158° in the crossbow planes at both x/L = 0.6 and x/L = 0.772. This is manifested by the conspic- uous dips in the measured Cp data shown in Figure 7. The predictions widely vary among the models. The SA model barely captures the fetaure, giving the shal- lowest dip and yielding the minimum pressure location more leeward than all the other models and the data. The KO-2 model seems to closely capture both the lo- cations and magnitudes of the minimum Cp. KO-2 also best reproduces Me Cp-plateau seen right after the pn- mary separation which was found in the experiments to occur near o = 123° and o = 112° at x/L = 0.6 and x/L = 0.772, respectively. Furthermore, atx/L = 0.772, the KO-2 model appears to capture, far closer than other models, the small kink caused by a secondary vortex above the surface, which was found to occur at o = 140° secondary _ / y separation line reattachment line windward AL 0.600 x/L = 0.772 primary separation line Fig. 6 Wall limiting streamlines showing the pattern of the crossflow separation at a = 20° - based on the KO-2 prediction (a) X/L = 0.600 -'~'~1I,I,,I,,I,, O Measured _ -- SA SST --- KO-1 KO-2 \ , , I , . 14/° . I . , I 04 ~ 0 30 60 90 120 150 180 1.', / ,, 1.~;' /.',`,o /, /.' (b) x/L = 0.772 0~ -0 1 . oC-0.2 -0.3 id- ~ ~ ~ P art''''`"''! an, I ~ ~ I i I ~ I O Measured SST --- KO-1 KO-2 W~ _ ~ : ~ -04 ,, 1, , I,,,,, I, , 1 . . 0 30 60 %. ¢r' ~Y 1! ~0_ 90 120 150 180 ¢(degree) Fig. 7 Surface pressure (Cp) distributions at a = 20° pre- dicted with the fine mesh 8
(a) x/L = 0.600 0.006 c: 0.004 0.002 OO 0.006 c: 0.004 0.002 OA it' 30 60 90 120 150 180 (b) x/L = 0.772 n 1 1 . , 1 , , 1 , , I , , 1 30 60 90 120 150 180 ¢(degree) c' Fig. 8 Skin-friction (Cf) distributions at cc= 20° pre- ~40 dieted with the fine mesh in the experiment. However, KO-2 model deviates from the data and other models' results on the windward side of the primary separation. It is interesting to see large differences among the three k-m models. Evidently, the low-Reynolds number modification in the KO-2 model seems to play a significant role in making a large differ- ence between the KO-1 and KO-2 model results. The SST model gives only marginally better results than the SA model, falling behind the KO-1 and KO-2 models. As discussed in the foregoing, the circumferential lo- cations of minimum Cf and the separation locations are correlated to each other. The experiments indicate that the minimum Cf locations are located consistently on the leeward side of the actual separation line. This can be seen in Figure 8. Note that the primary separation lines were observed at ~ = 123° and ~ = 112° in the crossbow planes at x/L = 0.6 and x/L = 0.772, respec- tively. Again, the predictions widely vary in terms of the minimum Cf locations, with the maximum difference of approximately 15° between the SA and the KO-2 mod- els. The predicted locations of the Cf minima are seen to shift windward gradually as one goes from SA, SST, KO-1, KO-2 models. It is also observed that KO-1 and KO-2 models predict the Cf minima (and separation) (a) x/L = 0.600 TV 20 O -20 40 , , 1~ 1 . . 1 , . 1 ~ 0 30 60 90 120 150 180 ¢(degree) 0 Measured - - SA SST --- KO-1 KO-2 , ,~ 1 , . 1 (b) x/L = 0.772 O Measured - - SA SST --- KO-1 KO-2 O . . a , , I , , 1 , , 1 , . 1 , , 1 u 30 60 90 120 150 180 ¢(degree) Fig. 9 Wall-shear angle (pw) distributions at a = 20° pre- dicted with the fine mesh a little too early in comparison to the data, which is largely consistent with the results shown in Figure 5. The distributions of predicted wall-shear angle shown in Figure 9 give us a measure of the strength (magni- tude) of the crossbow and the rate of convergence (or divergence) of the wall limiting streamlines near the separation or reattachment lines. The separation and reattachment lines are likely to manifest themselves as local peaks and zero-crossings in these plots. The com- parisons among the models are largely in line with what Cp and Cf distributions show. The crossbow becomes progressively stronger in the order of SA, SST, KO-1, and KO-2 models. Thus far, we have looked at the surface quantities predicted with the fine mesh without using wall func- tions. Now we will present the results obtained on the coarse mesh using the wall function approach. Fig- ure 10, Figure 11, and Figure 12 depict the distribu- tions of Cp, Cf. and Do at a = 20°, respectively, pre- dicted using the coarse (wall function) mesh. The use of coarse mesh and wall functions precluded the KO-2 model (k-co model with a low-Reynolds number modi- 9
(a) AL = 0.600 -0.2 o -0.1 -0.2 no O<- ' 1 ' ' 1 ' ' 1 ' ' I ' . O Measured - - - SST (wall fn.) -O. 1 _ ~ KO-1 (wall fn.) - - RSTM-1 (wall fn.) RSTM-2 (wall fn.) RSTM-3 (wall fn.) \ · ~ 1 ~ ' -0.3 _ \ i) so _ 0.002 _ 04 , . 1 ,, 1 , 1,, 1 ,, 1,, O. . ( ) 30 60 90 120 150 1{ ;0 0 (b) x/L = 0.772 . O Measured - - - SST (wall fn.) KO-1 (wall fn.) - - RSTM-1 (wall fn.) RSTM-2 (wall fn.) RSTM-3 (wall fn.) -04 ,, 1 ,, 1 ,, I,, 1 ,, 1,, 1 0 30 60 90 120 150 180 ¢(degree) Fig. 10 Surface pressure (Cp) distributions at a = 20° pre- dicted using wall functions fication) from the comparisons shown here. First of all, the results from the SST and KO-1 models show only marginal differences between the coarse (wall function) mesh results and the corresponding fine mesh results. Practically, this finding is quite significant, inasmuch as it gives a credential to the wall function approach and the results based thereupon. The RSTM-1 model, the baseline SMC, performs clearly better than the SST model in terms of capturing the local variations of the surface quantities. Although not shown here, it should be mentioned that the RSTM-1 model yields measur- able improvements over linear k-£ models which failed to capture the prominent features of the flow. How- ever, the KO- 1 model seems to outperform the RSTM-1 model. The RSTM-2 and RSTM-3 models are seen to bring remarkable improvements over the baseline RSTM model, outperforming the k-co models. It is quite interesting that seemingly minor changes in the c-equation lead to such disproportionately large differ- ences in the results. These improvements in the predic- tion of the surface quantities are directly carried over to the prediction of force and moment, as will be shown later. (a) x/L = 0.600 u.w- 0.006 0.004 to Into 0.006 0.004 0.002 Or~o I , , ~ , . ~ ~ ~ 1 ' ' I )~-'=~ ~ o Measured - - - SST (wall fn.) KO-1 (wall fn.) - - RSTM-1 (wall fn.) RSTM-2 (wall fn.) RSTM-3 (wall fn.) cry 30 60 90 120 150 180 (b) x/L = 0.772 O Measured - - - SST (wall fn.) KO-1 (wall fn.) - - RSTM-1 (wall fn.) RSTM-2 (wall fn.) RSTM-3 (wall fn.) JO 30 60 90 120 150 180 ¢(degree) Fig. 11 Skin-friction (Cf) distributions at a = 20° pre- dicted using wall functions Comparisons of the coarse mesh results shown here and with the fine mesh results presented earlier indicate that, insofar as high-Reynolds number flows are con- cerned, CFD predictions based on wall functions are perhaps better than has been commonly believed. A similar conclusion was made by Kim (2002) based on the numerical study of flow around a ship hull which shares some of the salient features with the present flow. Turbulence kinetic energy field The recent experiments by Chesnakas et al. (1998) and Goody et al.~2000) provide a wealth of turbu- lence data which include individual Reynolds-stresses, higher-order correlations, surface pressure fluctuations, and corresponding spectra. In this paper, we will discuss only the TKE field in the crossbow plane et x/e = 0.772 for the a = 20° case. The experimental data for the crossbow plane at x/L = 0.772 (Chesnakas and Simp- son, 1997) show that the peak value of TKE, 0.022pUo, occurs in the free-vortex sheet at 0 = 140° and at ap- proximately 30% of the local radius of the section from the body surface. The measurement also indicates that there is a region of relatively low TKE near 0 = 120° just downstream of the primary separation line occur- 0
(a) x/L = 0.600 O Measured - - - SST (wall fn.) KO-1 (wall fn.) - - RSTM-1 (wall fn.) RSTM-2 (wall fn.) RSTM-3 (wall fn.) \ 40 20 . -2() 20 _ ~4 5 lo _ -20 -40 , , 1 0 30 60 90 120 150 180 n no \ (b) x/L = 0.772 ~ it_ a I ~ ~ O o.os 0.1 0 15 o Measured I ° - - - SST (wall fn.) _ ~ KO-1 (wall fn.) - - RSTM-1 (wall fn.) RSTM-2 (wall fn.) RSTM-3 (wall fn.) 0.06 _ 0.04 x/L=0.772 KO-l -40` ~ 30 60 90 120 ¢(degree) 150 180 0.06 Fig. 12 Wall-shear angle (pw) distributions at a = 20° pre- ° ° dieted using wall functions nils ringato=115°. Figure 13 shows the contours of TKE (k/UO)at the crossbow plane predicted by all three k-m models with the fine mesh, whereas Figure 14 depicts the TKE at the same crossbow plane predicted by the three RSTMs with the coarse (wall function) mesh. The contours share some similarities among the results from differ- ent models, largely reflecting the formation of the free- vortex layer emanating from the body surface. Yet it can be seen that the k-m models yield contours that appear to be more spread out (diffused) than the RSTMs. Perhaps more important, all the models, except the ASTM-2, severely underpredict the peak TKE (0.022pUo). The location and magnitude of the peak TKE predicted by the RSTM-2 are remarkably close to what the experi- mental data indicate. Steady lift and pitching moment predictions The lift acting on slender bodies like prolate spheroids is characterized by a nonlinear increase of lift with incidence angle. The nonlinear lift is often called "vortex" lift because the augmented lift is due to the low pressure at the core of the vortices which x/~=n772 .;2 K0-2- o 0 0.05 0.1 0.15 n 0.1 >0.08 non 0.04 0.02 I_ 0.1 0.15 Fig. 13 Turbulent kinetic energy (k/Uo ) at x/L = 0.772 and a = 20° predicted by k-c,) models 11
0.12 0.1 0.08 0.06 0.04 0.02 . o' 1 0.05 0.14 0. 0.1 0.04 0.02 o 0 o.os 0.14 0.12 0.1 0.06 0.04 0.02 0 0 , 0 0.05 - x/L=0.772 RSTM-1 l - x/L=0.772 _ RSTM-2 z x/L=0.772 RSTM-3 z 0.15 Fig. 14 Turbulent kinetic energy (k/Uo) predictions at x/L = 0.772 and a = 20° predicted by RSTMs is impressed upon the near-by body surface. Accurate prediction of the location and strength of the vortices is therefore prerequisite to successful prediction of the force and moment. The predicted lift (Cr _ FN/O.SPUOL2) and pitching moment coefficients (CM _ Mz/o.5pUoL3) for the full range of incidence are plotted in Figure 15 along with the measured ones. The results shown in Figure 15 were obtained on the fine mesh. Like the results pre- sented so far, the predictions vary in a wide range. The differences among the models increase with the inci- dence. All models underpredict the lift. The SA and the KO-2 model yield the lowest and the highest lift, respec- tively. The lift predictions are largely consistent with the behaviors of the surface quantities discussed before. In- terestingly, the predicted pitching moment coefficients shows an opposite trend. The SA model matches closest with the measurement, while the KO-2 model deviates most from the data. Figure 16 depicts the result based on the coarse (wall function) mesh. It is noteworthy that the lift predictions by SST and KO-1 models based on wall functions are slightly higher than the corresponding results based on the fine mesh. However, the differences are deemed insignificant, insofar as we are comparing the results based on two drastically different near-wall modeling strategies. The RSTM-1 prediction is largely compara- ble to the SST and KO-1 model results. The RSTM-2 and RSTM-3 results are in commendable agreements with the data. Unsteady lift and moment characteristics for dynamic pitch-up motion The dynamic pitch-up maneuver has also been simu- lated by solving unsteady RANS (URANS) equations. However, the calculation was made using the SST model only for now. The plot on the right in Figure 17 depicts the time history of the lift as the incidence an- gle is dynamically increased from 0° to 30° in 11.0 dimensionless time units (t* - tUoo/L). The predicted unsteady lift was found to lower than the steady lift at the corresponding incidence angles, which is consistent with what the experimental data show. One interesting observation from this preliminary result is that the cal- culation appears to reproduce the transitory increase in the lift, which was persistently observed in the experi- ment (Wetzel and Simpson, 1997) in the early stage of the pitch-up motion, as reflected by a hump in Figure 17 during t* = 1.0 ~ 4.0. We surmise that this is caused by a starting vortex generated on the leeward side of the body by the impulsive start of the pitch-up motion, which contributes to augmenting the lift until it is swept away downstream by the flow. The computations with other turbulence models are reserved for future study. 12
(a) Steady lift force 0.o3ot 1 1 1 1 1 1 0.02C 0.02C V O.01< 0.01C 0.005 0.000` ) 5 10 15 20 Angle of attack (deg.) · · Measured _ o SA : ~ SST <I KO-1 v KO-2 / / ~ / ~ ~ 0 (b) Steady pitching moment 0.000 -0.002 Go -0.004 -0.006 -0.008 ~v / f ~ / 0 1 1 1 1 · · Measured 0 SA a\ i' SST \ v KO-2 \ ~ v e 1 1 1 1 1 1 ) 5 10 15 20 25 30 Angle of attack (deg.) Fig. 15 Force and moment predictions with the fine mesh (a) lift (CL) (b) pitching moment (CM) SUMMARY AND CONCLUSION The turbulent shear flow around a 6:1 prolate spheroid at the Reynolds number of 4 x 106 was stud- ied numerically using the Reynolds-averaged Navier- Stokes equations. The study covered the steady flow for a full range of incidence angle (a = 10° ~ 30°) and the unsteady flow associated a dynamic pitch-up ma- neuvenng. Both fine meshes and coarse meshes were used to investigate the effects of near-wall treatment. Several turbulence models, including Spalart-Allmaras' one-equation model, Menter's SST k-m model, Wilcox' k-m model, and three variants of Reynolds-stress trans- port model, were employed to evaluate the perfor- mances of the models for the present flow. The numer- ical results were compared with the experimental data in terms of several key aspects of the flow such as pat- tern of the crossbow separation, distributions of surface pressure, skin-frichon and wall-shear angle, and finally lift and pitching moment. The fidelity of the predictions varied in a wide range 0.03C 0.02C 0.02C V~0.014 0.01C o.ooC 25 30 0.00Go- u 0.000 _ -0.002 _ V-0.004 _ -0.006 _ -0.008, ~ 5 1 (a) Steady lift force · · Measured 0 SST (wall fn.) KO-1 (wall fn.) RSTM-1 (wall fn.) v RSTM-2 (wall fn.) o RSTM-3 (wall fn.) ,( <-a-! 1 ~ 5 10 15 20 Angle of attack (deg.) (b) Steady pitching moment ~ is. 1 1 25 30 1 1 1 1 1 | ~ · Measured 0 SST (wall fn.) KO-1 (wall fn.) RSTM-1 (wall fn.) v RSTM-2 (wall fn.) 0 RSTM-3 (wall fn.) I_ \ 11 ~ 10 15 20 25 30 Angle of attack (deg.) Fig. 16 Force and moment predictions with the coarse mesh (a) lift (CL) (b) pitching moment (CM) among the turbulence models. Some turbulence models employed here were found to be reasonably success- ful in predicting the salient features of the flow such as the separation patterns, wall pressure distribution, skin-friction distribution, lift force, and pitching mo- ment reasonable well. Among our significant findings are: · Mimicking the turbulence trigger numerically af- fects the predictions very little. · Some of the eddy-viscosity based models that have shown considerable promise in two-dimensional or quasi-two-dimensional flows were found to per- form rather poorly for the present flow. · Overall, the revised k-ce model of Wilcox (1998) seems to work better than other EVMs employed in this study. In particular, adding the low- Reynolds number effects (KO-2) makes measur- able improvements in all important aspects of the flow.
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