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The Flow Past a Wit Body Junction An Experimental Evaluation of Turbulence Models W. Devenport, R. Simpson (Virginia Polytechnic Institute and State University, USA) ABSTRACT Detailed three-component LDV measurements have been made in the flow of a turbulent boundary layer past an idealized wing-body junction. These measurements, which show great variety and three-dimensionality in the vortex- dominated turbulence structure of this flow, are here used to evaluate a number of turbulence models. Many of these models require or imply a relationship between the angles of the turbulence shear-stress and mean-velocity vectors. In the present flow these angles are not only different but do not follow any simple relationship. To predict the shear-stress angle, accurate modeling of the full shear-stress transport equations is clearly needed. In particular, new models based on measurements are needed for the pressure- strain term. The ability of six turbulence models to predict the magnitude of the shear-stress vector from the mean-velocity and/or turbulence kinetic energy is examined. Among the best are the Cebeci-Smith and algebraic-stress models. Other models, particularly the Johnson-King, are not well suited to this vertical flow. INTRODUCT ION This paper follows several (Davenport and Simpson, 1986, 1987, 1988a, 1988b, 1988c, 1990a) in which we have presented detailed velocity measurements made in the flow of a turbulent boundary layer around the nose of a wing-body junction. The purposes of this paper are; (i) to briefly review these measurements, (ii) to present new measurements made around the rest of the junction, and (iii) to use the whole data set to evaluate the usefulness and generality of a variety of turbulence models and modelling parameters. For a review of other experimental work on wing-body junction flows see Devenport and Simpson (1990a). EQU I PMENT Only abbreviated descriptions are given here; for complete details see Devenport and Simpson (1990b). The Wing and Wind Tunnel The wing (figure 1) is cylindrical, has a maximum thickness (T) of 71.7mm, a chord of 305mm and a span of 229mm. In cross section its shape (figure 2) consists of a 3:2 elliptical nose (major axis aligned with the chord) and a NACA 0020 tail joined at the maximum thickness. Trips are attached to both sides of the wing to ensure steady and fixed transition. The wing is mounted at zero sweep and incidence at the center of the flat 0.91-m wide test wall of the Virginia Tech Boundary Layer Tunnel, forming the junction. In the absence of the wing this tunnel produces a flow of zero streamwise pressure gradient, consisting of a closely uniform (to within 1%) low turbulence (0.2%) free stream and an equilibrium two- dimensional turbulent boundary layer (see Ahn (1986)) on the test wall. With the wing in place, inserts attached to the wind tunnel side walls are used to minimize blockage-induced pressure gradients. TV optics: U. v system B ~~~.~! ~ ~ .,` .. x,u LDV optics: i/,/\\ 0, W systemic B Figure 1. Perspective view of junction. Laser Doppler Velocimeter (LDV) AA unshifted BB -IS MHz shirts CC +21.5 I\1Hz shirtea the wing-body A 3-component LDV was used to measure detailed profiles of mean-velocity and turbulence quantities in 6 planes surrounding the wing. These planes (numbered 1,3,4,5,8, and 10 for organizational reasons) are illustrated in figure 2. The LDV has three sets of sending optics, two of which are shown schematically in figure 1. W.J. Devenport and R.L. Simpson, ACE Dept., VPI&SU, 215 Randolph Hall, Blacksburg VA 24061 815

- 2 -1 .5 -.5 ~ O .5 1 1.5 2 Each set produces an arrangement of beams sensitive to a different pair of velocity components and their associated Reynolds shear stress. Only one set is used at a time. The flow is seeded using dioctal phthalate smoke (typical particle diameter 1 micron). Light scattered from the measurement volume is focussed onto the pinhole of a single photomultiplier tube. Data are obtained from the photomultiplier signal using either fast sweep rate sampling spectrum analysis (see Simpson and Barr, 1975) or a DANTEC 55N10 Burst Spectrum Analyzer. Velocity statistics are obtained by time (not particle) averaging and thus should be free of bias. Measurements presented here have been corrected, where necessary, for velocity gradient broadening and finite transit time broadening using the techniques described by Durst et al. (1981). Uncertainty estimates for 95% confidence limits are listed in table 1. . Uncertainty +.03 Uref ~1 Plane 8 Plane 5 Plane 3 _ l l - -2 -1 0 Plane 10 o O ~' ~ '''t 5tO // ,°.~;~ :,~L~ 1 ,. 4 ~ - 1~. ~`~ . \ ~ , C'/'j' 1~`'~"'' - ~: {! ~ // . - , , , , ~ ~ ~- _ 5:o 5~° ~1 ° o O ~ O 'AN o; /of ~ ~ ~\ 1, 1 ,,, 1 , , , , 1 , 1 1 1 1 2 3 4 X / T Figure 2. Contours of mean-surface pressure coefficient Cp on the wall surrounding the wing, 0 - locations of LDV profiles, line of separation, ~ "line of low shear, ~~. locus of peak turbulence kinetic energy in the vortex. Quantity Mean velocity Turbulence kinetic energy, k Turbulence _ear- stress -uv Turbulence _ear- stress -vw Turbulence shear- stress magnitude Table 1 Typical uncertainties measurements. 95% confidence limits. 1 1 1 COORDINATE SYSTEMS, TEST CONDITIONS +.00033 Uref2 +.00027 Uref2 Most results and discussion will use the lab fixed coordinate system X,Y,Z and U,V,W centered at the intersection of the wing leading edge and wall (figure 1). X is measured downstream from the leading edge, Y normal to the wall and Z completes a right-handed system. In presenting LDV measurements the additional coordinate S will be used. S is measured along any of the LDV measurement planes from the wing surface or flow centerline, as shown in figure 2. In discussing turbulence models and parameters other coordinate systems will be used distinguished by subscripts. Subscripts 'f', 's' and 'g' refer to coordinates fixed in the local free-stream direction, the local mean-flow direction and the local direction of the mean- velocity gradient vector, respectively. In calculating these directions V component velocities will be ignored. In all coordinate systems, upper case and lower case symbols will be used to denote the mean and fluctuating components of velocity respectively. Distances will in general be non + 00026 U. ' dimensionlized on the maximum thickness of the - ref wing (T), velocities on the undisturbed approach free-stream velocity, Uref. Under nominal test conditions the momentum th~ckness Reynolds number of the approach boundary layer, measured in the plane of symmetry 2.15T upstream of the wing leading edge, was 6700, corresponding to a total boundary layer thickne~s ~ of 36mm (.50T) and Urf of 27 m/s. e +.00033 Uref2 in LDV 816

EXPERIMENTAL RESULTS Figure 2 shows contours of mean surface pressure coefficient C (based on undisturbed free-stream conditions) Pand principle features of an oil-flow visualization performed on the wall surrounding the wing. Figures 3 and 4 show m~an- velocity vectors and contours of turbulence kinetic energy k/Uref2 measured in planes 1 through 10. The mean-velocity vectors represent components normal to the centerline of the horseshoe vortex defined as the locus of peak turbulence kinetic energy (see figure 2). Other projections of the mean-velocity field (e.g. normal to the wing, parallel to the measurement planes) do not clearly show the secondary-flow velocities associated with the vortex. Note that the measurements presented here in planes 1, 3 and 4 have previously been published by Devenport and Simpson (1987, 1988a, 1988b and 1990a). This flow is dominated by the pressure field produced by the wing and the velocity field generated by the horseshoe vortex that is wrapped around the junction between the wing and wall. In the plane of symmetry upstream of the wing (plane 1) the oncoming boundary layer experiences an adverse pressure gradient that causes it to separate 0.47T upstream of the leading edge (figure 2). The separation region formed (figure 3(a)) is dominated by the recirculation associated with the horseshoe vortex. This roughly elliptical structure, centered at X/T = - .2, Y/T = .05, generates an intense backflow by reversing fluid impinging on the leading edge of the wing. The backflow reaches a maximum mean velocity of -0.48Uref and then decelerates, giving the appearance of reseparation between X/T = -.25 and -.3. Reseparation, however, does not occur as a thin region of weak reversed flow is sustained adjacent to the wall. This region is then all that remains of the backflow upstream to the separation point. The near reseparation of the backflow produces a distinct line in the surface oil-flow visualization known as the line of low shear (figure 2). In the vicinity of the horseshoe vortex the turbulence stresses (and thus the turbulence kinetic energy) become very large reaching values an order of magnitude greater than in the approach boundary layer (figure 4(a)). These large stresses are associated with bimodal (double-peaked) histograms of velocity fluctuations like those shown in figure 5, and are produced by intense low-frequency bistable unsteadiness in the structure of the vortex. This unsteadiness is a result of the turbulent/non- turbulent intermittence of fluid entrained into the corner between the wing and wall (Davenport and Simpson (199Oa)). Moving out of the plane of symmetry, fluid experiences a strong favorable pressure gradient (figure 2) that accelerates''~as it moves around the nose. Close to the wing in planes 3, 4 and 5 this acceleration, acting in concert with the rotational motion of the vortex (which here is bringing low-turbulence high-momentum fluid from the free-stream down close to the wall), locally relaminarizes the boundary layer (Davenport and Simpson (1988b and c)). Turbulence shear stresses in this region are therefore much smaller than elsewhere. Turbulence kinetic energy (figures 4(b), (c) and (d)) is also reduced. Although the intensity of turbulent fluctuations in the vicinity of the vortex falls in the favorable pressure gradient the peak values of turbulence kinetic energy remain many times those in the surrounding boundary layer because of the bimodal unsteadiness. Bimodal histograms are seen in the vicinity of the vortex in planes 3, 4 and 5 (figures 4(b), 4(c) and 4(d)). Despite the favorable pressure gradient the vortex clearly grows in this region moving away from the wall and the wing (figures 3 and 4). (Projected onto i _ Y/ T _ 0.0024 _ ~ ('~'°~1 1 0.0136 ~ ._ 11; ~ 1 )! .~ 0.0031 11 0.0159 ~ 0.0789 -0.0038 Ill 0.01946 - 0.0963 0.00~- 0.02~ ~ 7, 0.0292 0.1434 . ~ ~ J ~ 0.0357 0.1 756 ,fL 0.0432 0.'147 - 1 0.()527 0.0644 0.0963 [~ 0.1175 - / \ 1 n 1414 ~ 1756 0 3 -3 0 3 -3 0 3 _, 1,u2.' Figure 5. Histograms of U-component velocity fluctuations measured at X/T = -.2 in plane 1. the wall the centerline of the vortex fairly closely follows the line of low shear, see figure 2.) In addition mean secondary flow velocities fall by a factor of about 2 between planes 1 and 5 (figure 3). Downstream of the maximum thickness this flow is subjected to an adverse pressure gradient (figure 2) that appears to cause rapid growth in the vortex and a dissipation of the bimodal unsteadiness. (Note the change in scales between different parts of figures 3 and 4.) Bimodal histograms were not observed in planes 8 and 10 and peak turbulence kinetic energies are much lower here than upstream. Secondary-flow velocities, which are also reduced in the adverse pressure gradient, become much more difficult to 817

.Z5 . 15 .05 C 35 4 L , L_ - . 3 ~1- c ~: ~ ~- .2 L , ~ ~ t .cs ~_ 0 ~ I .4 ~ _ .35 _ .3 _ ., . 25 _ -;_ ~ .15 _ .1 _ _ ns .6 (b) Plane 3 \ _ ~ ~ \ \ _ _ ~ ~ \ \ ~ ~ ,-~a,,,~',j,~,,T, 5 .4 .3 .2 .1 0 S / T Uret (c) Plane 4 Figure 3. Mean secondary flow field generated by the vortex. .5 .4 .3 .2 .1 0 S / T Ure ~2 (a) Plane 1 . ~s ~. .05 _ ~.. v :! ~ I .... I -.6 - .5 - .4 - .3 - .2 X / T (a) Plane 1 ~_ :~ 4 0 (b) Plane 3 .4 .35 .3 .2S .2 . 15 . ~ .05 o o ~.~ .6 .5 .4 .3 .2 _ S / T (c) Plane 4 1 n Figure2 4. Contours of turbulence kinetic energy k/Uref in the vicinity of the vortex. Dotted lines enclose the regions in which bimodal histograms are observed 818

: ~ Is t .4 ~ r~ -~ -_ . B _ 7 _ .6 _ .8 ,4 .2 - (f) Plane 10 Figure 3. Mean secondary flow field generated by the vortex. 819 . .s _ r . _ _ , ~ . _ ~ ~9 (d) Plane 5 (e) Plane 8 _ -_^ -_ - ~ ~ \ ~ ~ N.. w..~...L,..: _ .3 .2 .1 S / T ~ o .2 o .8 .7 .6 .5 .4 S / T (d) Plane 5 .3 .2 . ~O i: l.2 1 .B .6 .4 .c S / T (e) Plane 8 0 .8 .6 . 4 .2 o (f) Plane 10 .6 .4 .2 0 Figure 4. Contours of turbulence kinetic energy k/Uref2 in the vicinity of the vortex. Dotted lines enclose the regions in which bimodal histograms are observed

distinguish from the rest of the mean-velocity field (figures 3(e) and (f)). Despite these changes the region between the vortex and the wing remains one of low turbulence shear-stresses because of the free-stream fluid entrained here by the vortex. Figures 3 and 4 represent only a fraction of the mean-velocity and turbulence information we have collected. All mean-velocity and Reynolds stress components, some triple products and histograms of fluctuations in all three components have been measured at over 1400 points in this flow. The quantity of experimental data and the variety of turbulence structure in this flow make it, in our opinion, ideal for testing the generality and therefore usefulness of turbulence models. EVALUATION OF TURBULENCE MODELS General Remarks Before evaluating the validity of turbulence models it is appropriate to discuss the relationship between the turbulent shear stress and velocity gradient directions since many models use or imply such a relationship. The shear-stress and velocity-gradient vectors are defined as having components -uv, -vw and OU/3Y, aW/BY in the X and Z directions respectively. Their directions are given by the angles, a~tan~~( w) and ag=tan~~(aWu/3Y) (1) Most often the shear-stress and velocity gradient angles are assumed to be the same, i.e. - vw ~W/dY - uv - vw or -uv au/ay au/aY aw/aY A" shown this implies that the streamwise and cross-flow eddy viscosities are the same. Although this is ideal for converting turbulence models designed for two-dimensional flows to three dimensions, it is not supported by the present or past experiments (see Johnston (1970), van den Berg and Elsenaar (1972), Fernholz (1981) and others). Figure 6 shows a plot of spanwise vs. streamwise eddy viscosity for all points inside the boundary layer in planes 3 through 10. Points outside the line of separation, where the direct effects of the horseshoe vortex and its bimodal unsteadiness are much smaller, are plotted with different symbols to those inside. In neither region does there appear to be any significant correlation between these two parameters. A possible improvement has been suggested by Rotta (1977) who derives an alternative relationship between the eddy viscosities using the transport equations for the shear stresses approximated for thin shear layers, D(-UV) _v2 U_p/ (aUy+ax)+,~3 (~;71+UV2) (3) Dt -v aY- p ( aY+ aZ ) + By ( p +wv ) (4) The terms from left to right represent convection, production, pressure strain and diffusion. By substituting the Poisson equation for the fluctuating pressure p' it can be shown that the pressure strain is composed of two terms, the first ~1 is associated with the interaction of the mean strain and fluctuating velocities and the second ~2 with the interaction of the fluctuating velocities alone. .2 is usually 004 .,~ ~n o ~n a, 3 o 1 CO o- 002 c~ 002 o -.004 . -o - o o c o O o OO~b ,` . ~. ~ ; o o c: n 1 , , , 1 , , ,.W , , , ~ , , . . . - .004 - .002 0 .002 .004 Streamwise eddy v i scos i ty Figure 6. Spanwise vs. streamwise eddy viscosity in local mean flow coordinates for all me ~urement points inside the boundary layer (Vu /U ~ 5%). Squares represent points inside the line of separation. approximated by the shear stress itself multiplied by a factor related to the turbulence kinetic energy (see Rodi (1984)), that factor being the same in both equations. Rotta approximated .1 using the Poisson equation for the pressure fluctuation p' and by assuming local symmetry in the turbulence structure. Neglecting convection and diffusion, which can be shown to be higher order terms for thin shear layers, and dividing equation (2) by equation (1), he then obtained the expression, tan(a ~-a) - T.tan(ag-a) - V~Ws - u,svg ( 5 ) or / - T. / i.e. the cross-flow eddy viscosity is an empirical constant T times the spanwise eddy viscosity, in local flow coordinates. Unfortunately, as can be seen from figure 6 this equation is no more valid than equation (2) in the present flow. This result is confirmed by figure 7 in which values of T deduced from these measurements are plotted together as a histogram. This shows a large spread with T varying over a range of at least +2. We have tested a number of other hypothetical relationships between the shear- stress and velocity gradient angles also without success. These have included a relationship between at ~ a and the local cross-flow velocity Wf/Ue, one bet9ween the spanwise and streamwise eddy viscosities, and one between at and a9 based on van den Berg's (1982) hypothesis. There are two principle reasons for the failure of the above concepts. The first becomes apparent if we transform the problem to coordinates based on the direction of the local mean velocity gradient vector (subscript 'g'). In this system the cross-flow shear stress exactly represents the lag or lead of the angle of the shear stress vector over that of the velocity gradient vector, i.e. _g ~ tan(a~-ag) (6) -u`~v 820

qn so 20 0 o -4 -3 -2 ~R~T! _ ~ =:t':~ 0 1 2 3 4 l Figure 7. Histogram of values of Rotta's T parameter compiled from all--measurement points inside the boundary layer (Ju2/U > 5%). Also in this system, however, the transport equation for the~ross-flow stress (equation (4)) looses the term v dW/8Y since by definition dW/BY is zero. The cross-flow stress and the lag of the shear-stress vector are therefore determined entirely by the unknown pressure strain and the neglected convection ~ d diffusion terms which in the absence of v2aW/aY are likely to be important. The second reason for the failure is the way in which the pressure strain terms are usually modelled. It is simple to show that (without the boundary-layer approximation) the pressure strain and pressure diffusion can be combined into a single term with the form, . . . ~ WAX +u Spy (7) in the uv transport equation, and -v ,a~Z +w pappy (8) in the vw transport equation. These terms obviously cannot be modelled by substituting for B~'/6X, Bp'/3Y and Bp'/8Z from the Navier-Stokes equations or an approximation to them (as done by van den Berg (1982)) since this will lead to an where identity or an expression of the error in the approximation. By the same argument, substituting for p' using the Poisson equation (which is just the Navier Stokes rearranged) or an approximation (as is done in effect by Rotta (1977)) must also eventually lead to an identity or an expression of the error in the approximation. Neither of and these approaches are therefore valid. The pressure strain terms can in general only be modelled by substituting different moments of Navier Stokes equations and/or by using information derived from experiments. In our opinion only the latter approach is likely to prove successful since higher moments of the Navier-Stokes will only introduce more unknowns. Careful experiments in which the pressure strain terms are (presumably) measured by difference are therefore needed. Turbulence models In this section the assumptions of several prescribed eddy viscosity models, the k-e model an algebraic-stress model and Bradshaw's (1971) model are tested. For each model, predictions of the magnitude of the shear-ntress vector from the measured mean-velocity or turbulence kinetic energy distributions are compared with measurements. For models that use the eddy- viscosity at the shear-stress magnitude is assumed to be given by 1pl ~ (~j2+~2)~ Vt[( aaU)2+( ~aW)2] 2 (9) i.e. the cross-flow and streamwise eddy viscosities are assumed equal. Although the results of the previous section show that this is not the case, there appears to be no better alternative. Since skin-friction data are not yet available for the present flow the wall treatments employed by most of the models are not tested and are ignored in the following discussion. Comparisons with experimental data do not include points in the near-wall region Y/T < .02 (y+ less than about 120). The authors concede that several of the models examined here were never intended for use in flows as complex as this one. However, they, or models like them, are often used in complex flows. It is therefore important that their limitations be known. In the first and simplest turbulence model considered here the eddy viscosity is prescribed as a function of Y entirely in terms of mixing length '1', 1 - BY Y/6 < A/~ 1 ~ 18 Y/8 2 A/x where the eddy viscosity is given by vie - 12[( BU)2+( aW)2] ~ (10) ( 11) A and K are empirical constants and ~ is the boundary-layer thickness. From two-dimensional test calculations Patankar and Spalding (1970) suggest A = .09 and K ( the van Karman constant) - .435. The Cebeci-Smith and Johnston-King turbulence models, described for three- dimensional flows by Abid (1988), are variations on this basic form. The Cebeci-Smith model is described by the relations V ~ - V t (1 - exp(-v~i/v`O)) (12) V ~ 12~( ~U)2~( ~W)2] 2 (13) 1 - lCY, 1C - . 4 a V t - O . 0168ykll(Q.-p)dyl (14) o Yk ~ [1+5.5(~)6] 1 (15) Q jU2+w2 Qua - Qly-a (16) the principal difference with the basic model being the explicit prescription of the eddy viscosity in the outer region in terms of the Klebanoff intermittence function Yk and the use of a smoothing function between the inner and outer regions. The Johnson-King model uses the same smoothing function but def ines 821

V ~ ~ 1 ( _ ) 2 1 - Icy, tic - . 4 and where v t - 0.ol680ykli(0e-Q)dyI (18) o p m ~( ~;2 +~;j2 ) 2 I m ( 19 ) the maximum turbulence shear stress in the profile. The Johnson-King model was originally designed for two-dimensional adverse pressure gradient and separated boundary layers in which the maximum shear stress appears to be an appropriate scaling parameter. In the turbulence model Am is determined from a differential transport equation. The parameter ~ is chosen so that the equation am as V [ ( )2+ ( ey) 2] (20) is satisfied at the location of maximum shear stress in each profile. This requires an iterative procedure. The above three models were used to calculate the turbulence shear-stress magnitude from the measured mean-velocity field. In the case of the Johnson-King model the maximum shear stress and its location were also provided from the experimental data. The k-e model is one of the most widely used in calculating two-and three-dimensional turbulent flows. Coupled with the wall treatment of Chen and Patel (1988) it has been used by Deng (1990) to calculate the flow past a wing-body junction very similar to that studied in the present experiments. The k-e model defines the eddy viscosity in terms of the turbulence kinetic energy k and the dissipation c, k2 V t ~ Cp - (21) k and ~ are determined from approximate transport equations (see Rodi (1984) or Abid and Schmitt (1984)), Dk _ - Uv aU _ ~ aw ~ + any E ( v + o t ) sky ] a [( ~' aft (23) (22) The empirical constants are usually given the values, C~-O.O9, ok-1, Ce1-1 57, 24 CC2-2 O. a~ 1.3 ( ) The ~ equation could not be tested using the present measurements. The k equation was tested by substituting the eddy-viscosity and the velocity gradients for the Reynolds stresses and substituting equation (21) for the dissipation. This gives, (17) Dt ~ v`[t~y)2~(~y)2]_ ~(25) + aids+ V t) Ok] Using the measured distributions of k and the mean-velocity gradients this equation was solved iteratively for vt. Initial and boundary values for at required for this calculation were determined from the measurements. Convection of k normal to the LDV measurement planes was ignored in this calculation since it could not be deduced from the measurements. This term was almost certainly negligible at most points. The algebraic stress model uses transport equations for k and ~ similar to those above. However, instead of relating the turbulent stresses to k and ~ through an eddy viscosity, algebraic equations for the individual stresses are used. These are derived from the full (differential) stress transport equations by assuming, among other things, that the convection and diffusion of the individual stresses is proportional to that of k. The algebraic stress equations, written in standard tensor notation in their full form (Rodi (1984), Abid and Schmitt (1984)), are En and. Cl+P/~-l where Pj. is the production of ujuj, P is the production of k, djj is 1 of i=j and zero otherwise, and ~iw is a component of the pressure-strain correlation that accounts for wall proximity effects. ~iw is an algebraic function of only k, a, ~ and the stresses themselves (see Abid and Schmitt (1984)). The following values for the empirical constants, suggested by Abid and Schmitt (1984) and Launder (1982), were used y-0.55, C1-2.2, C/-0.5, ~-0.3 (27) C'1 and C' appearing in the equations for Id Equation (26), together with the definition of of, give seven algebraic equations for the six Reynolds stresses and ~ in terms of k and the mean-velocity field. If the latter are provided from experimental data then the stresses and ~ can be deduced. This requires an iterative Newton-Raphson procedure since the equations are non-linear. Combining uv and vw then gives the magnitude of the shear stress. In performing this calculation production terms associated with gradients of V and W normal to the measurement planes were ignored since they could not be obtained from the measurements. These terms were almost certainly negligible at most points. Note that equation (26) does not involve boundary layer approximations. Without these the algebraic stress model can, at least in theory, predict a lag or lead in the angle of the turbulence shear- stress vector. Bradshaw' s ( 1971 ) turbulence model for three-dimensional boundary layers uses approximate dif ferential transport equations for uv and vw. These are derived by analogy with the transport equation for k assuming a simple constant of proportionality between k and the shear-stress magnitude, a tu~+vw2) 2 (28) 822

By analogy with two-dimensional flows Bradshaw suggests a value of 0.15 for al. Bradshaw's model was tested simply by multiplying measured values of k by 0.15 to obtain estimates of the shear stress magnitude. Results ~ turbulence model calculations are presented in figures 8 and 9. Figures 8(a) through 8(i) show measured and computed contours of shear-stress magnitude in plane 8 located towards the trailing edge of the wing (figure 2). Because of space limitations detailed comparisons in other planes are not presented. However, figures 9(b) through 9(i) show, for each turbulence model, histograms of the ratio of computed to measured shear-stress magnitude compiled from data in all planes. For reference figure 9(a) shows a probable histogram of experimental error in the measured values of the shear-stress deduced from uncertainty estimates. Note that figures 9(a) through 9(g) do not contain data from close to th~-wall (Y/T < .02) or from in the free stream (Vu2/U ~ 5%). Of the prescribed eddy-viscosity models (figures 8 and 9 (b) through (e)) the Cebeci Smith appears to be the best. Although there are some obvious qualitative differences in the shapes of the measured and computed shear-stress contours in plane 8 (compare figures 8(a) and (d)) these do not represent large quantitative differences at most points. In other planes there are some large differences, however, as indicated by the histogram in figure 9(d). According to this histogram the r.m.s. error in predictions with the Cebeci-Smith model is about 70% while the mean error is only +3%. Clearly the worst of these three models is the Johnson-King which produces an unrealistic shear-stress field in the vortex (figure 8(e)). This model fails because the eddy viscosity distribution it prescribes depends not just on the peak shear-stress magnitude but also, implicitly, on the distance from the wall at which it occurs. Moving across plane 8, or any other plane through the vortex, the peak jumps from the near-wall region to the center of the vortex producing a sudden and unrealistic change in the prescribed eddy-viscosity profile. There are also problems with the smoothing function used in this model, equation (12). This function requires at ~ vj at the maximum shear-stress location, a condition not always met near the center of the vortex. Note that the histogram of calculated to measured shear-stress magnitude for the Johnson-King model (figure 9(e)) is misleading since it, and the mean and r.m.s. errors stated on it, do not include many points where the computed shear-stress magnitude exceeded 4 times that measured. Also, the peak near 1 in this histogram does not necessarily represent any accuracy in the model since the maximum shear-stress magnitude and its location were provided to the model from the experimental data. The model is therefore bound to produce accurate estimates of the shear-stress magnitude at and near this point. Although the shear-stress magnitude distributions produced by the basic mixing length model of equations (10) and (11) appear qualitatively realistic (figure 8(b)), the histogram in this case (figure 9(b)) shows large quantitative discrepancies (mean and r.m.s. errors +57% and 97% respectively). A detailed comparison with the measurements shows that most of the larger discrepancies occur in the near- wall region where the mixing length is prescribed a" a linear function of Y with slope K = 0. 435 (equation (10)). This suggests that a different value of K might improve the predictions. Figures 8(c) and 9(c) show that some improvement is achieved by optimizing this constant to 0.3. The Results for the r.m.s. error with ~ = 0.3 is still 86% however. Unlike the above models the k-e, algebraic stress and Bradshaw's (1971) model were given the measured distribution of turbulence kinetic energy from which to calculate the shear-stress magnitude. It is surprising then that these models appear to perform little better (see figures 8 and 9 (f) through (i)). The k-e model does not accurately reproduce the features of the shear-stress field in plane 8 (figure 8(f)), the vortex being much flatter and the point of maximum shear-stress magnitude occurring much closer to the wing than in the measurements. As indicated by the histogram (figure 9(f)) the k-e model also does poorly in the other planes, the mean and r.m.s. errors in its predictions being +58% and 83% respectively. Bradshaw's model produces slightly better qualitative agreement in plane 8 but over estimates the shear-stress magnitude at most locations (figure 8(h)). This model seems unable to account for the low shear stress levels in the region adjacent to the wing where the vortex is bringing low-turbulence fluid down close to the wall. As shown by the histogram (figure 9(h)) the shear-strens magnitude is also over-predicted in other planes, the mean and r.m.s. errors being +63% and 72% respectively. These errors can be reduced somewhat by optimizing the value of a (see figures 8(i) and 9(i)). A value of 0.12 seems to best suit the present data set reducing the r.m.s. error to 68%. Much of this remaining error results from over-estimation of the shear stress magnitude in the low turbulence region between the vortex and wing, which still persists . The algebraic stress model, perhaps the best of these three, produces the most realistic shear-stress contours in plane 8 (figure 8(g)) especially away from the wall. Overall comparisons with shear-stress measurements in other planes (figure 9(g)) give mean and r.m.s. errors of +12% and 75%. Theoretically, at least, the algebraic stress model is capable of predicting leads and lags in the angle of the shear-stress vector relative to the mean-velocity gradient vector. As shown in figure 10, lag angle predictions appear much smaller than, and largely uncorrelated with, measured angles. 1 5 & u In O 1 4~ - - 5 - 1 n ~ 0 0 ~ lo lo lo lo -1 - 5 1 o lo O 0~ lo ,,,, [ ] i n lo tan(at ~ as) measured 0 lo To $oo° ] or ~ ~ o o o to lo lo lo 5 1 Figure 10. Comparison of angles between the turbulence shear-stress and mean-velocity gradient vectors measured and computed using the algebraic stress model. 823

3 7 6 5 4 3 2 a . .7 . B _ .6 1.2 1 .B .6 .4 .2 S / T (c) Basic mixing-length model ~ = .3 Figure 8. Contours of measured and computed turbulence shear-stress magnitude V(~+~)/Uref2 in plane 8. 7000 6000 5000 4000 \ 1 2000 _ ~ 1000 ,,7 ~_-~ . ~ t P in - Irk V) to D z _ ~ ~64~ \ ~ ~ l 2 - ~ ,. =~: V V t . 2 ~. ~ . 6 . 4 2 ~a (b) Basic mixing-length model ~ = .435 _. _ _ .. . . . .. Il!,e~l - into Standard dev~ati0~4 = 0 0q7 0 1~ t. ........ lo .s 1 15 2 2.5 3 ., ~- 3~ ~ (a) Measurements (a) Probable histogram of error in shear-stress magnitude measurements. ~- Mean = I .56 Standard deviation = 0q7 (b) Basic mixing-length model K = .435 Mean = I . 21 Jard deviation = ~ 86 2 ] 4 I_ 3 3.5 4 (c) Basic mixing-length model K = . 3 Figure 9. Histograms of the ratio of computed to measured turbulence shear stress magnitude compiled from co baritone at points inside the boundary layer (Vu /U > 5%) but outside the near- wall region (Y/T > .02). 824

- . B _ .6 _ .5 _ S . .6 .5 .2 . ~ (d) Cebeci Smith model ~ D .e (e) Johnson-King model B _ .7 .6 _ .5 .4 1.2 1 .8 5 / T .2 (f) k-e model . 1 v Figure 8. Contours of measured and cO2mputed2 turbulence shear-stress magnitude ](uv +vw )/Uref in plane 8. 80 70 60 50 4( 2( (d) Cebeci Smith model 60 I 50 i 1 40 cat us TO 30 a, A 20 10 n ~ 1 o ~ _ 60 50 in - Q ~ 40 U7 to ~ 30 a, D go 20 10 a Mean - 1.03 Standard deviation = (570 l 11 Hi r 2.5 3 3.5 4 Meall = 1.12 Standard deviation = C)-7z rT ~ r..~ ~ 1.5 2 2.5 3 3.5 4 (e) Johnson-King model l Mean = ~ .58 Standard deviation = 0-83 1 l 1 ill 0 .5 (f) k-e model Figure 9. Histograms of the ratio of computed to measured turbulence shear stress magnitude compiled from comparisons at points inside the boundary layer (VuZ/U > 5%) but outside the near- wall region (Y/T > .02). 825

. B .7 6 .5 . 4 . 3 .2 . ~ B .6 _ .5 _ ~ ' _ 50 7 t 4 _ U 1 1.2 Me~n = 1.12 1 n(larn nP ~'~ t I nn = n 7'i \ ~ s -~ G: ~_=4 'w~ ~ ~ ~ ~,~,3 1 .6 5 / T .2 ~o 5 1 1.5 2 2.5 3 3.5 4 (g) Algebraic stress model v, q) a, - . 4 (h) Bradshaw's (1971) model, a1 = .15. B . 7 .6: .5 _ . 4 _ > _ .3 . ~ o V 0 ~.5 60 50 40 ~0 20 10 O v, - u) ~ ~ ~0 ~ z 1.2 1 .8 .6 ._ S / T 90 ~0 70 60 50 40 30 20 10 o (i) Bradshaw's (1971) model, a1 = .11. (i) Figure 8. Contours of measured and cO2mputed2 turbulence shear-stress magnitude V(uv +vw )/Uref in plane 8. (g) Algebraic stress model Me~n = I .62 StandaPt deviation = 07 ~h~ 3 3.5 4 (h) Bradshaw's (1971) model, a1 = .15. . Mean = I .29 i Standard deviation = 068 .5 1 .~.~ 2 2.5 3 3.5 4 Bradshaw's (1971) model, a1 = .11.. Figure 9. Histograms of the ratio of computed to measured turbulence shear stress magnitude compiled from co ~arisons at points inside the boundary layer (Vu /U > 5%) but outside the near- wall region (Y/T > .02). 826

In summary, it appears that more complex turbulence models do not necessarily do better than simpler ones. It could be argued that, out of the 6 models tested here, the Cebeci Smith is the best. Despite the fact that the Cebeci Smith uses a prescribed eddy viscosity profile intended for much simpler flows than the present, its predictions of shear-stress magnitude from the mean-velocity field alone are on the whole better than those of the more complex models. This implies, contrary to conventional thinking, that the more complex models are no more general. CONCLUSIONS New velocity measurements, made in the flow past a wing body junction, have been presented. Combined with earlier results these show the formation and development of the horseshoe vortex and its three-dimensional turbulence structure around the entire wing. A number of turbulence models have been examined using these data. Many of these models require or imply a relationship between the angles of the turbulence shear-stress and mean- velocity vectors. In the present flow these angles are not only different but do not follow any simple relationship. To predict the shear- stress angle, accurate modeling of the full shear-stress transport equations is clearly needed. In particular, new models based on measurements are needed for the pressure-strain term. The assumptions of several prescribed eddy viscosity models, the k-e model, an algebraic- stress model and Bradshaw's (1971) model have been tested. For each model, predictions of the magnitude of the shear-stress vector from the measured mean-velocity or turbulence kinetic energy distributions were compared with measurements. The Cebeci-Smith eddy-viscosity model is among the best. The algebraic stress and Bradshaw's model also do well but appear to gain little from their relative complexity. Other models, particularly the Johnson-King, are not well suited to this vertical flow. All the experimental data presented and referred to here is available in tables and on magnetic disc from the authors. ACKNOWLEDGEMENTS The authors would like to thank Dr. S. Olamen and Mr. A. Obst for their help in taking some of the above measurements. This work was sponsored by NAVSEA through NSWC contract N00014- 87-K-0421. REFERENCES Abid R. 1988, "Extension of the Johnson-King turbulence model to the 3-D flows", AIAA paper 88-0223, 26th Aerospace Sciences Meeting, Reno, Nevada, January 11-14. Abid R and Schmitt R. 1984, "Critical examination of turbulence models for a separated three dimensional turbulent boundary layer", Rech. Aerosp., No 6. Ahn S. 1986, "Unsteady Features of Turbulent Boundary Layers", M.S. Thesis, Dept. of Aerospace and Ocean Engineering, VPI&SU. Bradshaw PP, 1971, "Calculation of three- dimensional turbulent boundary layers", Journal of Fluid Mechanics, vol.46, p.417-445. Chen H C and Patel V C, 1988, "Near-wall turbulence models for complex flows including separation", AIAA Journal, vol. 26, p.641-648. Deng G. 1989, "Resolution des equations Navier Stokes tridimensionelles. Application au calcul d'un raccord plaque plane-alla", PhD thesis, Universite de Nantes, France. Devenport W J and Simpson R L, 1990a, t'Time- dependent and time-averaged turbulence structure near the nose of an wing-body junction", Journal of Fluid Mechanics, vol. 210, pp 23-55. Devenport W J and Simpson R L, 1990b, "An experimental investigation of the flow past an idealized wing-body junction: preliminary data report", AOE Dept., VPI&SU. Devenport W J and Simpson R L, 1988a, "The turbulence structure near an appendage-body junction", 17th Symposium on Naval Hydrodynamics, The Hague, The Netherlands. Devenport W J and Simpson R L, 1988b, "LDV measurements in the flow past a wing-body junction", 4th International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal. Devenport W J and Simpson R L, 1988c, "Time- dependent structure in wing-body junction flows", Turbulent Shear Flows 6, Springer Verlag. Devenport W J and Simpson R L, 1987, "Turbulence structure near the nose of a wing-body junction", AIAA paper 87-1310, AIAA 19th Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, Hawaii. Devenport W J and Simpson R L, 1986, "Some time- dependent features of turbulent appendage-body juncture flows", 16th Symposium on Naval Hydrodynamics, Berkeley, California. Durst F, Melling and Whitelaw J H, 1981, Princinles and Practice of Laser Do~pler Anemometry, NY: Academic Press. Fernholz H H and Vagt J D, 1981, "Turbulence measurements in an adverse pressure gradient three dimensional turbulent boundary layer along a circular cylinder", Journal of Fluid Mechanics, vol. 111, p233. Johnston J P, 1970, "Measurements in a three- dimensional turbulent boundary layer induced by a swept forward-facing step", Journal of Fluid Mechanics, vol. 42, p823. Launder B E, 1982, "A generalized algebraic stress transport hypothesis", AIAA Journal, vol. 20, p. 436-437. Patankar S V and Spalding D B, 1970, Heat and Mass Transfer in Boundary Layers, Second edition, Intertext, London. Rodi W, 1984, Turbulence Models and Their Application in Hydraulics - A State of the Art Review, Second Edition, IAHR, Delft, The Netherlands. Rotta J C, 1977, "A family of turbulence models for three-dimensional thin shear layers", Symposium on Turbulent Shear Flows, University Park, PA. Simpson R L and Barr P W, 1975, "Laser Doppler Velocimeter Signal Processing Using Sampling Spectrum Analysis", Rev. Sci. Inst., 46, pp. 835- 837. van den Berg B, 1982, "Some notes on three- dimensional turbulent boundary-layer data and turbulence modelling", Three-Dimensional Turbulent BoundarY Lavers, p. 1-18, Springer. 827

van den Berg B and Elsenaar A, 1972, "Measurements in a three-dimensional incompressible turbulent boundary layer in an adverse pressure gradient under inf inite swept wing conditions", NLR-TR-72092U. DISCUSSION Fred Stern The University of Iowa, USA Is there a reason that the Baldwin-Lomax turbulence model was not chosen for evaluation? As I'm sure you are aware, this model is the workhorse of the aerospace industry and also used extensively by the Navy laboratories and others. Also, was it possible through the comparisons to reach any conclusion with regard to the quasi-steady assumption, which is made in most current turbulence models? AUTHORS' REPLY The Baldwin-Lomax model is identical to the Cebeci-Smith model in the inner region of an attached two~imensional turbulent boundary layer. The form of the outer region model is also identical to the Cebeci-Smith model, with the differing length scales. Stock and Haase, AIAA Journal, Vol. 27, pp. 5-14, 1989, show that the Cebeci-Smith model performs better than the Baldwin-Lomax model for the two-dimensional flows tested. 828