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LES of turbulent flow past a swept fence
International Journal of Heat and Fluid Flow 24 (2003) 606–615 www.elsevier.com/locate/ijhff
LES of turbulent flow past a swept fence L. di Mare *, W.P. Jones Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London SW7 2AX, UK Received 23 November 2002; accepted 15 March 2003
Abstract The present study deals with the large-eddy simulation of the span-wise invariant turbulent flow past a swept fence at low Reynolds numbers. The swept fence geometry was introduced by McCluskey et al. [Eighth Symposium on Turbulent Shear Flows, 1991, p. 9.5.1] as a way of gathering information on the behaviour of near wall flows beneath a separation region in the presence of a significant cross flow. The configuration avoids both the lack of generality of typical statistically two-dimensional flows and the difficulties of fully three-dimensional flows. In the present case, large-eddy simulation is both a challenging and promising approach. On the one hand, as in most low-Reynolds number and recirculating flows, simple wall models cannot be applied [Engineering Turbulence Modelling and Experiments 2, Florence, Italy, p. 303] on the other hand, due to the relatively low Reynolds of the flow and the type of scaling suggested in Hardman and Hancock [Exp. Fluids 27 (2000) 653], accurate resolution of the near wall region can be achieved without incurring prohibitively high computational costs. A variant of the dynamic SGS model of Germano et al. [Phys. Fluids A 3 (1991) 1760] with a smooth and reliable numerical behaviour is also tested in this flow. The agreement with the experimental data of Hardman [Moderately three-dimensional separated and reattaching turbulent flow. Ph.D. thesis, University of Surrey, 1998] is found to be satisfactory. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: LES; Separation; SGS modelling; Spanwise invariance; Swept fence; Turbulence
1. Introduction Experimental and numerical studies of statistically two-dimensional separated flows are numerous and have led to a number of well established results. It is well known that boundary layer properties and state upstream of separation have a considerable effect on the the structure of such flows (Adams and Johnston, 1985; Davies and Snell, 1977). The issue of the turbulence structure in separated two-dimensional flows or in the shear layers bounding them have also been addressed by Castro and Haque (1987). Quantitative results are also available in the form of SimpsonÕs law for the near-wall flow beneath a separated region (Simpson, 1983), which represents for separated boundary layers what the loglaw represents for attached boundary layers. ‘‘Universal’’ scalings are also known to hold for separated flows originating from thin boundary layers and with no im*
Corresponding author. Tel.: +44-20-7594-7037; fax: +44-20-75815495. E-mail address: [email protected] (L. di Mare).
posed pressure gradient (Smits, 1982). Detailed numerical studies of separated flows have been performed over the years by a host of researchers. To cite but a few we recall the successful computations of flow past a backward-facing step by DNS (Le et al., 1997), and by LES (Akselvoll and Moin, 1997). Most separated boundary layers of practical interest are three-dimensional in nature and extremely complex. One of the first systematic experimental studies of threedimensional turbulent recirculating flows was performed by McCluskey et al. (1991), who also proposed a classification of mildly three-dimensional recirculating flows as span-wise-invariant (type A), diverging (type B) and converging (type C). These flows are especially suitable for theoretical analysis and experimental and numerical studies because at most four out of nine velocity gradient components are non-vanishing, either because of invariance or because of symmetry. Detailed measurements have been performed in recent years at the University of Surrey in mildly three-dimensional converging (Hardman and Hancock, 2000) and span-wise-invariant flows (Hancock, 1999). To date only a few numerical
0142-727X/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0142-727X(03)00054-7
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Nomenclature C C Cð~Þ Cf Cp Gðx; x0 Þ hf L Rehf s T uc u; v; w Uref U0 W0
model parameter provisional estimate C C at the test filter level friction coefficient pressure coefficient filter function fence height Leonard stress tensor Reynolds number Re hf ¼ ðUref hf =mÞ strain tensor sij ¼ 12 ouj =oxi þ oui =oxj test filter level SGS stress tensor outflow convection velocity velocity components free-stream velocity (magnitude) streamwise free-stream velocity U0 ¼ Uref cos # spanwise free-stream velocity W0 ¼ Uref sin #
results are available for the flow configurations considered in the present work. RANS results are presented by Hardman (1998), while the results of a DNS of flow past a swept backward-facing step have recently been reported by Kaltenbach and Janke (2000). Numerical studies of statistically two-dimensional fence flows have been performed by Orellano and Wengle (2000) and Pascarelli et al. (2001). The present paper presents LES results obtained in conditions matching the experiments performed by Hardman (1998), i.e. relating to flow regions of type A, in the separated turbulent flow past a swept fence. In the target experiments the free stream velocity is 5.8 m/s, and the fence, placed at the front of a splitter plate, is 1 cm high and 3 mm thick. The Reynolds number based on free stream velocity U0 and fence height hf is Reh ¼ 3900. The sweep angle is 10°. The attachment length XA from Hardman (1998) is 21 hf ; XA is referred to as attachment rather than reattachment length because the separating streamline does not coincide with the attaching streamline, as happens in statistically twodimensional flows (McCluskey et al., 1991). In the following sections the mathematical model is examined, with particular attention to a novel formulation of the SGS model, results are presented in Section 3 and finally conclusions are drawn in Section 4.
x; y; z; t coordinate axis and time attachment length XA Greeks D # m q r s
filter width dissipation scale sweep angle kinematic viscosity density viscous stress tensor SGS stress tensor
Super and subscripts i; j; k indices for the coordinate axis – filtered field ~ test filtered field a anisotropic part of a tensor
axis orthogonal to the fence and the z axis parallel to it. The origin of the reference frame lies at the base of the fence, as shown in Fig. 1. LES is based on an approach whereby only the largest and most energetic scales of motions are directly computed, whilst small scales of motions are modeled (Smagorinsky, 1967). More precisely, any given function f is decomposed according to: Z f ¼ f fsgs ; f ¼ Gðx; x0 Þf ðx0 Þ dx0 ð1Þ R3
The filter function G is a suitably localized function, so that f only retains the variability of f over length scales comparable or larger than the filter width D. Though several choices are possible in the present work a Ôtop hatÕ filter: 1 for jxi x0i j < D2 ; 3 0 Gðx; x Þ ¼ D ð2Þ 0 otherwise: is used (Germano, 1992), as it fits naturally into a finite volume formulation.
2. Mathematical model 2.1. LES approach For the purpose of the present computations we use the fence frame of McCluskey et al. (1991), with the x
Fig. 1. Computational domain and relevant parameters.
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Applying the decomposition (1) to the Navier–Stokes equations, for incompressible flows with constant properties, under the hypotheses that filtering and differentiation in space commute, gives rise to: oui o ui o p=q o uj þ ¼ ð rij þ sij Þ oxi oxj ot oxj
ð3Þ
ouj ¼0 oxj
ð4Þ
In Eq. (3), rij ¼ 2msij is the viscous stress tensor, while sij ¼ ui uj ui uj represents the effect of the SGS motion on the resolved motion and is known as the SGS stress. The SGS stress must be modeled before the filtered equations can be solved. The continuity equation (4) remains unchanged on application of the filter, due to its linear nature. 2.2. Sub-grid scale model The model for the SGS stress used in the present work is the dynamic model of Germano et al. (1991) implemented using the approximate localization procedure of Piomelli and Liu (1995) together with a modification proposed by the authors. This model represents the SGS stresses as the product of an SGS viscosity mSGS times the resolved part of the strain tensor. The SGS viscosity is evaluated as the product of the filter length D times an appropriate velocity scale. The appropriate choice for this velocity scale is: Dksk. Thus, for the anisotropic part of the SGS stresses the following expression is found: saij ¼ 2ðCDÞ2 ksksaij
ð5Þ
where the model parameter C needs to be estimated. In the dynamic model this is achieved by applying a second filtering operation, denoted by ~, to Eq. (3). In the test filtered equation the SGS stresses are: Tij ¼ ug uj ui ~ i uj ~
ð6Þ
Comparing the definition of Tij with s~ij we obtain uj ug uj u~i ~ Lij ¼ Tij s~ij ¼ i
ð7Þ
Eq. (7), known as GermanoÕs identity (Germano et al., 1991), involves only known, i.e. resolved, quantities. In order for Eq. (7) to yield the needed expression for C we need to assume some form of relationship between the model constant values C and Cð~Þ at the grid- and testfilter levels. Based on the hypothesis that the cutoff length falls inside the inertial subrange, the approximation conventionally used is: C2 ¼ C2 ð~Þ
ð8Þ
However, such a subrange is not always guaranteed to occur in low-Reynolds number or wall-bounded flows. In particular it is expected that the largest deviations
from universality of the SGS motions will occur in the regions of weakest resolved strain. As a consequence the two values of the model parameter at two different filter levels are likely to differ. In order to account for this it is therefore proposed that: ! 2 2 ð9Þ C ð~Þ ¼ C 1 þ pffiffiffi 2 2 2D k~skk~sa k2 The parameter has the dimensions of dissipation and, if the flow has only one length scale ‘ and one velocity scale v, the following estimate can be used: v3 ‘ though a more refined estimate is probably desirable. With v ¼ U0 and ‘ ¼ hf for the present flow we obtain ¼ 2:02 104 m2 s3 . The reasoning leading to Eq. (9) is that the scale invariance of C can only be invoked if the cutoff falls inside an inertial subrange. When this happens the dissipation provided by the model should represent the entire dissipation present in the flow. On the other hand in the high Reynolds number limit the dissipation is only determined by v and ‘ (with the proviso that the turbulence can be characterised by single length and velocity scales), so that the ratio of to D 2 k~sk3 is a measure of how far the flow is from scale preserving conditions. Eq. (9) represents a first order expansion of the scaledependent expressions for C such as the one proposed by Porte-Agel et al. (2000) using a single length and velocity scale. Making use of Eqs. (7) and (9) yields, on contraction of both sides with the tensor ~s: pffiffiffi 2 ½2 2ðC2 DÞ ksksaij ~ ~saij Laij~saij 2 pffiffiffi 2 C ¼ ð10Þ 2 þ 2 2D k~skk~sa k
where C2 is a provisional value for the field C2 , e.g. its value at the previous time step (Piomelli and Liu, 1995). The dependence embodied in Eq. (9) yields a simple expression for C2 and its evaluation requires only slight modifications of the approximate localization procedure of Piomelli and Liu (1995). Furthermore the formulation is well conditioned and avoids the spiky and irregular behavior exhibited by some other implementations of the dynamic model. It is also to be noted that when the resolved strain tends to zero, C2 also tends to zero, whilst C2 ð~Þ remains bounded. This situation is encountered in the computation of the free stream of external flows. Expression (10) yields smooth C2 fields with no need for averaging. The maxima of C2 thus evaluated are of the same order of magnitude as LillyÕs estimate for the Smagorinsky model constant Lilly (1967), and are located in zones of strong resolved shear, where in low-Reynolds number flow some dissipation is likely to occur at grid filter level. Very small values are
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found in regions where the flow is smooth. The use of Eq. (9) in Eq. (7) amounts to adding a significant diagonal contribution to the integral operator in Eq. (7) thereby improving its conditioning, hence the smooth behavior of C when evaluated according to (10). Eq. (7) provides three independent equations for C2 that cannot be simultaneously satisfied. Contraction of both sides of saij amounts to solving for C2 in a least Eq. (7) with ~ square sense. However, Eq. (10) does not prevent the appearance of negative values of the model parameter, and these values have to be set to zero, in order to prevent instability. Negative values of the SGS viscosity are found on roughly 12% of the grid points at each time step and on these points the SGS viscosity mSGS is set to zero. In the implementation of the model test-filtering is performed in all space directions, and no averaging is performed of the computed model parameter field. The ratio D~=D is set to 2 and the filter width is determined from D ¼ ðDx Dy Dz Þ1=3 . 2.3. Numerical procedure and computational parameters All the computations have been performed with the in house code BOFFIN (Jones, 1991). The code has previously been employed by the authors in the LES of turbulent flow past a backward facing step at Reh ¼ 5100 (di Mare and Jones, 2001), with results in satisfactory agreement with the DNS of Le et al. (1997) and the LES results of Akselvoll and Moin (1997). The code has also been used in the LES of a jet in cross flow (Jones and Wille, 1996a,b), and of vortex shedding past a square cylinder (di Mare and Jones, 1998). The code implements an implicit finite-volume incompressible flow solver using a co-located variable storage arrangement. Because of the co-located arrangement fourth
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order pressure smoothing, derived from that proposed by Rhie and Chow (1983), is applied in order to prevent spurious oscillations of the velocity field. Time advancement is performed via an implicit Gear method for all transport terms. The overall procedure is second order accurate in both space and time. The time step is chosen by requiring that the maximum Courant number lies between 0.1 and 0.3. This requirement is enforced for reasons of accuracy (Choi and Moin, 1994). The code is parallel and uses the message passing interface MPI-1.2. Computations have been conducted on the Cray T3E-1200 at CSAR, Manchester, using 16, 32, 128 and 512 processors. More details on the numerical algorithm and its implementation are available in di Mare and Jones (2001), Jones and Wille (1996a) and di Mare and Jones (1998) and references cited therein. Several combinations of grids and boundary conditions have been tried. The domain and grid sizes for all the test cases are shown in Table 1, together with grid spacings in wall-units, based on the maximum friction coefficient from the experiments (0.002). The computations with free-slip upper boundary, A and B, give attachment lengths XA ¼ 15:6 hf and 17.3 hf , respectively, compared with the experimental value of XA ¼ 21 hf , the better value coming from the more resolved computation. The computations with entraining upper boundary, C, D, E and F give XA ¼ 17:6 hf , 24 hf , 18.5 hf and 21.8 hf , respectively, as shown in Table 1. The domains used for runs C and D are very narrow and are most to affect the computed results. For this reason runs D and E have been performed. These two runs have the same grid, twice as wide as runs C and D, better resolved in the span-wise and stream-wise directions, but with the same resolution in the wall-normal direction. In run E the dissipation parameter has been evaluated as ¼ 0:001v3 =l while in run D is evaluated as ¼ v3 =l.
Table 1 Mesh parameters max Dxþ i
min Dxþ i
Expansion ratio
XA
4.90 4.40
1.28 1.16 1
15.6
0.52
1 1.04 1
17.3
67.1 135.0 34.1
1.45
1 1.05 1
17.6
59.1 29.1 3.39
33.8 136.0 34.1
1.41
1 1.04 1
24.0
41.4 29.1 6.19
24.6 136.0 10.4
1.41
1 1.04 1
18.5/21.8
Case
Direction
Points
Li =hf
A
x y z
204 60 80
50.2 7.69 10.0
87.3 59.0 20.7
x y z
282 148 84
40.3 13.8 6.38
23.2 72.2 13.6
x y z
204 116 21
58.5 29.1 3.39
x y z
404 116 21
x y z
388 116 116
B
C
D
E/F
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The differences in the computed results from tests E and F show that the model is indeed active also at high resolutions and low Reynolds numbers. Time averages are computed as integrals in time. Statistics have been accumulated for at least 300 hf =U0 for all test cases. The inflow boundary conditions are represented with a uniform velocity U0 . No perturbations are imposed. This choice is justified by the very low level ( 0.3%) of free-stream turbulence found in the target experiments of Hardman (1998). The inflow conditions are applied 10 hf upstream of the fence. No numerical oscillations are observed upstream of the fence. The outflow boundary is treated as a convective outflow (Pauley et al., 1990), i.e. oui oui ¼ uc ot ox
ð11Þ
where the convection velocity uc is the local time-average velocity vector evaluated at the outflow plane. The upper boundary is treated as a free-slip, non-porous wall for test cases A and B, as an entraining boundary for test cases C and D. The entraining boundary is represented with an homogeneous Neumann boundary for all velocity components and as a Dirichlet boundary condition for the pressure. The pressure field is the one corresponding to the potential flow in the free stream, i.e. is assumed uniform: oui ¼0 on
ð12Þ
p¼0
ð13Þ
Due to the large effect of the blocking ratio on the experimental results of Hardman (1998), a very high computational domain is used. The side boundaries are periodic. No-slip conditions are applied at wall boundaries and no wall-model is used. The lower boundary of the inflow section is also treated as a non-porous, freeslip wall.
Fig. 2. Instantaneous contours of SGS dissipation (a) and entrophy (b) on the centre-plane. Only part of the computational domain is shown.
of the SGS dissipation can be seen to occur across the shear layer bounding the separated flow and again across the boundary layer developing downstream of attachment. No significant dissipation occurs near the walls, nor, of course, in the undisturbed flow. The average behaviour of the model is shown in Figs. 3 and 4. More precisely Fig. 3 shows profiles of the time averaged model parameter C2 at several locations downstream of the fence. The highest values of C2 occur close to the obstacle, and decay quite rapidly at downstream locations. It is to be noted that the average value of C2 is always much smaller than the commonly used values of the Smagorinsky parameter, even though values similar to those reported are indeed attained instantaneously. The computed average of the SGS uvReynolds stress is shown in Fig. 4. Consistently with the behaviour of C2 , the SGS Reynolds stresses are highest close to the fence and then decay rapidly further downstream. Figs. 3 and 4 also show the effect of the dissipation parameter on the model parameter and SGS stresses. More precisely, in both runs D and F is evaluated as ¼ v3 =‘ and the model parameter C and the SGS stresses from the two runs are comparable, while in run E ¼ 0:001v3 =‘ and the C has values comparable with LillyÕs estimate throughout the boundary layer, except near the wall. Interestingly the SGS stresses from run E do not differ from those obtained from runs D and F as dramatically as the model parameters do. 3.2. Instantaneous flow field
3. Results and discussion 3.1. Model and SGS stresses Results concerning the time-dependent behaviour of the SGS model, Eq. (10) are shown in Fig. 2. Fig. 2(a) shows the instantaneous contours of SGS dissipation in the midplane. The high values of SGS are clustered across the shear layer bounding the separated flow region and, to a lesser extent, across the developing boundary layer downstream of attachment. The dissipation contours follow very closely the pattern of the resolved entrophy contours, shown in Fig. 2(b), which is consistent with the discussion in Section 2.2. Almost all
More results concerning the time-dependent behaviour of the flow field for this configuration are shown in Fig. 5. Fig. 5(a) presents instantaneous stream-wise velocity contours near the splitter plate at y=hf ¼ 0:09. The different structure of the flow field beneath the separated flow and beneath the developing boundary layer downstream of attachment are clearly visible. No well defined ‘‘streaks’’ are found in the flow even far downstream of attachment, indicating that the flow structure has not evolved to that of an attached boundary layer even by two attachment lengths downstream of the fence. No such structure is discernible in the separated flow region either. Some interesting observation can be
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Fig. 3. SGS model parameter at several locations downstream of the fence. - - -: run D, : run E, ––: run F.
Fig. 4. SGS uv Reynolds stress at several locations downstream of the fence. - - -: run D, : run E, ––: run F.
Fig. 5. Stream-wise (a) and span-wise (b) velocity contours near the splitter plate. Dark contours: positive values, light contours negative values, black contours zero.
made with the respect to the behaviour of the span-wise velocity component close to the wall. Beneath the separated region and close to the wall, the average value of the span-wise velocity must be positive, yet no dominant span-wise flow direction can be deduced from Fig. 5(b). The fact that the instantaneous span-wise velocity component bears very little resemblance to its mean pattern suggests that the span-wise flow in the configuration of interest is highly unsteady; consistent with the observations of Hardman (1998).
3.3. Flow statistics Fig. 6 shows the pressure and friction coefficients for test cases D, E and F compared to the experimental results (Hardman, 1998). The agreement is quite good in the separated region, especially for x=XA < 0:7. Some discrepancies can be observed in the recovery region. The friction coefficient in the z direction, Cfz also appears too low. The agreement in the separated flow region appears to be better for runs E and F, and it is
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Fig. 6. Pressure and friction coefficients. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
worse for run E. Further downstream all runs exhibit too low a Cfz : this behaviour is still being investigated. The average profiles of U =U0 and W =W0 are shown in Figs. 7 and 8, together with the measurements by Hardman (1998). The agreement for the u component is very good at all stations, while the w profiles are reasonably well reproduced, though some features are missing in the computational results, especially near the
wall, consistently with the results for the friction coefficients. Fig. 9 shows a comparison of the LES results from test cases D, E and F with SimpsonÕs law (Simpson, 1983), for separated flows far from attachment points.
y
y U
ð14Þ ¼A log 1 1 UN N N
Fig. 7. Average u-component velocity profiles at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
Fig. 8. Average w-component velocity profiles at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
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Fig. 9. Average u-component velocity profiles at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
where UN is the maximum negative velocity and N is the distance from the wall where it is achieved. A is a universal parameter with a commonly accepted value of A ¼ 3. The LES data exhibit a logarithmic region between y=N ¼ 0:02 and 0.2 that agrees with Simpson law for distances from the fence between 0:25XA and 0:75XA , while the profiles at 0:125XA and 0:875XA , as expected, do not follow the logarithmic law. The discrepancies between runs E and F show that the effect of the model in early stages of separation is to change the shape of the separated region and hence the position of the maximum negative velocity. Figs. 10–13 show turbulence intensities and resolved Reynolds stress profiles at several stations downstream of the obstacle. Here again agreement with the experimental results for run D is reasonable in the first
half of the separated region and worsens further downstream. The under prediction of the Reynolds stress is consistent with the under prediction of Cp and Cfx in the recovery region. Here the narrow domain precludes the representation of large structures that contribute significantly to the Reynolds stresses. This effect is not important upstream of attachment because there are no such large structures there, as the boundary layer at separation is laminar and the eddy structures have not grown to such an extent as to saturate the computational domain. This goes someway to explain the previous statement regarding Cp and Cfx . That this diagnosis is correct is supported by the better agreement between experiments and LES data for the Reynolds stresses and turbulence intensities for runs E and F.
Fig. 10. u-component turbulence intensity at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
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Fig. 11. v-component turbulence intensity at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
Fig. 12. w-component turbulence intensity at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
Fig. 13. uv Reynolds stress several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
4. Conclusions Some results from the LES of the spanwise invariant flow past a swept fence have been presented. The in-
stantaneous and time-average behaviour of the flow field is correctly reproduced by the method at a reasonably low computational expense. The paper has proposed a variant of the dynamic SGS model of Germano et al.
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(1991) which appears to be well suited to comparatively low-Reynolds number computations. The model is based on the possibility of allowing different values of the Smagorinsky constant at different filter levels. The model parameter is numerically well behaved and does not need averaging of the computed fields, thus making its extension to complex geometries straightforward and computationally attractive. Consistent with the experiments, the computations have been found to be very sensitive to spurious blocking effects, so that the best results have been obtained with a very high domain and entraining boundaries. Future work in the flow configuration will involve a better estimate for the dissipation scale in the proposed sub-grid model. Further computations are currently being run with a wider domain in the spanwise direction in order to remove the confinement effect shown in the results currently available.
Acknowledgements The authors are grateful to Dr. P.E. Hancock for providing experimental data and for helpful discussions. The authors also acknowledge support from EPSRC under grant GR/N17683.
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Germano, M., 1992. Turbulence, the filtering approach. Journal of Fluid Mechanics 238, 325–336. Germano, M., Piomelli, U., Moin, P., Cabot, W., 1991. A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A 3, 1760– 1765. Hancock, P.E., 1999. Measurements of mean and fluctuating wall shear stress beneath spanwise invariant separation bubbles. Experiments in Fluids 27, 53–59. Hardman, J.R., 1998. Moderately three-dimensional separated and reattaching turbulent flow. Ph.D. thesis, University of Surrey. Hardman, J.R., Hancock, P.E., 2000. The near-wall layer beneath a moderately converging three-dimensional turbulent separated and reattaching flow. European Journal of Mechanics B––Fluids 19, 653–672. Jones, W., Wille, M., 1996a. Large-eddy simulation of a plane jet in a cross-flow. International Journal of Heat and Fluid Flow 17 (3), 296–306. Jones, W., Wille, M., 1996b. Large-eddy simulation of a round jet in a cross-flow. In: Engineering Turbulence Modelling and Experiments 2. Jones, W.P., 1991. BOFFIN: a computer program for flow and combustion in complex geometries. Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine. Kaltenbach, H.J., Janke, G., 2000. Direct numerical simulation of flow separation behind a swept, rearward-facing step at ReH ¼ 3000. Physics of Fluids 12, 2320–2320. Le, H., Moin, P., Kim, J., 1997. Direct numerical simulation of turbulent flow past a backward facing step. Journal of Fluid Mechanics 330, 349–374. Lilly, D.K., 1967. The representation of small scale turbulence in numerical simulation experiments. In: Proc. IBM Scientific computing Symposium on Environmental Sciences, pp. 195–210. McCluskey, P., Hancock, P.E., Castro, I.P., 1991. Three-dimensional separated flows. In: 8th Symposium on Turbulent Shear Flows, pp. 9.5.1–9.5.6. Orellano, A., Wengle, H., 2000. Numerical simulation (DNS and LES) of manipulated turbulent boundary layer flow over a surfacemounted fence. European Journal of Mechanics, B––Fluids 19, 765–788. Pascarelli, A., Marra, F., Piomelli, U., 2001. Large-eddy simulation of the flow over a wall-mounted fence. In: Direct and Large Eddy Simulation IV. Kluwer Academic Publishers, pp. 419–426. Pauley, L.L., Moin, P., Reynolds, W.C., 1990. The structure of twodimensional separation. Journal of Fluid Mechanics 220, 397–411. Piomelli, U., Liu, J., 1995. LES of rotating channel flow using a localised dynamic model. Physics of Fluids 7 (4), 839–848. Porte-Agel, F., Menevau, C., Parlange, M., 2000. A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer. Journal of Fluid Mechanics 415, 261– 284. Rhie, C.M., Chow, W.L., 1983. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 21 (11), 1525–1532. Simpson, R.L., 1983. A model for the backflow mean-velocity profile. AIAA Journal 21 (1), 142–143. Smagorinsky, J., 1967. General circulation experiments with the primitive equations. Monthly Weather Review 91, 99–164. Smits, A.J., 1982. Scaling parameters for time-averaged separation bubble. Transactions of the ASME, Journal of Fluids Engineering 104, 178–184.
LES of turbulent flow past a swept fence L. di Mare *, W.P. Jones Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London SW7 2AX, UK Received 23 November 2002; accepted 15 March 2003
Abstract The present study deals with the large-eddy simulation of the span-wise invariant turbulent flow past a swept fence at low Reynolds numbers. The swept fence geometry was introduced by McCluskey et al. [Eighth Symposium on Turbulent Shear Flows, 1991, p. 9.5.1] as a way of gathering information on the behaviour of near wall flows beneath a separation region in the presence of a significant cross flow. The configuration avoids both the lack of generality of typical statistically two-dimensional flows and the difficulties of fully three-dimensional flows. In the present case, large-eddy simulation is both a challenging and promising approach. On the one hand, as in most low-Reynolds number and recirculating flows, simple wall models cannot be applied [Engineering Turbulence Modelling and Experiments 2, Florence, Italy, p. 303] on the other hand, due to the relatively low Reynolds of the flow and the type of scaling suggested in Hardman and Hancock [Exp. Fluids 27 (2000) 653], accurate resolution of the near wall region can be achieved without incurring prohibitively high computational costs. A variant of the dynamic SGS model of Germano et al. [Phys. Fluids A 3 (1991) 1760] with a smooth and reliable numerical behaviour is also tested in this flow. The agreement with the experimental data of Hardman [Moderately three-dimensional separated and reattaching turbulent flow. Ph.D. thesis, University of Surrey, 1998] is found to be satisfactory. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: LES; Separation; SGS modelling; Spanwise invariance; Swept fence; Turbulence
1. Introduction Experimental and numerical studies of statistically two-dimensional separated flows are numerous and have led to a number of well established results. It is well known that boundary layer properties and state upstream of separation have a considerable effect on the the structure of such flows (Adams and Johnston, 1985; Davies and Snell, 1977). The issue of the turbulence structure in separated two-dimensional flows or in the shear layers bounding them have also been addressed by Castro and Haque (1987). Quantitative results are also available in the form of SimpsonÕs law for the near-wall flow beneath a separated region (Simpson, 1983), which represents for separated boundary layers what the loglaw represents for attached boundary layers. ‘‘Universal’’ scalings are also known to hold for separated flows originating from thin boundary layers and with no im*
Corresponding author. Tel.: +44-20-7594-7037; fax: +44-20-75815495. E-mail address: [email protected] (L. di Mare).
posed pressure gradient (Smits, 1982). Detailed numerical studies of separated flows have been performed over the years by a host of researchers. To cite but a few we recall the successful computations of flow past a backward-facing step by DNS (Le et al., 1997), and by LES (Akselvoll and Moin, 1997). Most separated boundary layers of practical interest are three-dimensional in nature and extremely complex. One of the first systematic experimental studies of threedimensional turbulent recirculating flows was performed by McCluskey et al. (1991), who also proposed a classification of mildly three-dimensional recirculating flows as span-wise-invariant (type A), diverging (type B) and converging (type C). These flows are especially suitable for theoretical analysis and experimental and numerical studies because at most four out of nine velocity gradient components are non-vanishing, either because of invariance or because of symmetry. Detailed measurements have been performed in recent years at the University of Surrey in mildly three-dimensional converging (Hardman and Hancock, 2000) and span-wise-invariant flows (Hancock, 1999). To date only a few numerical
0142-727X/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0142-727X(03)00054-7
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Nomenclature C C Cð~Þ Cf Cp Gðx; x0 Þ hf L Rehf s T uc u; v; w Uref U0 W0
model parameter provisional estimate C C at the test filter level friction coefficient pressure coefficient filter function fence height Leonard stress tensor Reynolds number Re hf ¼ ðUref hf =mÞ strain tensor sij ¼ 12 ouj =oxi þ oui =oxj test filter level SGS stress tensor outflow convection velocity velocity components free-stream velocity (magnitude) streamwise free-stream velocity U0 ¼ Uref cos # spanwise free-stream velocity W0 ¼ Uref sin #
results are available for the flow configurations considered in the present work. RANS results are presented by Hardman (1998), while the results of a DNS of flow past a swept backward-facing step have recently been reported by Kaltenbach and Janke (2000). Numerical studies of statistically two-dimensional fence flows have been performed by Orellano and Wengle (2000) and Pascarelli et al. (2001). The present paper presents LES results obtained in conditions matching the experiments performed by Hardman (1998), i.e. relating to flow regions of type A, in the separated turbulent flow past a swept fence. In the target experiments the free stream velocity is 5.8 m/s, and the fence, placed at the front of a splitter plate, is 1 cm high and 3 mm thick. The Reynolds number based on free stream velocity U0 and fence height hf is Reh ¼ 3900. The sweep angle is 10°. The attachment length XA from Hardman (1998) is 21 hf ; XA is referred to as attachment rather than reattachment length because the separating streamline does not coincide with the attaching streamline, as happens in statistically twodimensional flows (McCluskey et al., 1991). In the following sections the mathematical model is examined, with particular attention to a novel formulation of the SGS model, results are presented in Section 3 and finally conclusions are drawn in Section 4.
x; y; z; t coordinate axis and time attachment length XA Greeks D # m q r s
filter width dissipation scale sweep angle kinematic viscosity density viscous stress tensor SGS stress tensor
Super and subscripts i; j; k indices for the coordinate axis – filtered field ~ test filtered field a anisotropic part of a tensor
axis orthogonal to the fence and the z axis parallel to it. The origin of the reference frame lies at the base of the fence, as shown in Fig. 1. LES is based on an approach whereby only the largest and most energetic scales of motions are directly computed, whilst small scales of motions are modeled (Smagorinsky, 1967). More precisely, any given function f is decomposed according to: Z f ¼ f fsgs ; f ¼ Gðx; x0 Þf ðx0 Þ dx0 ð1Þ R3
The filter function G is a suitably localized function, so that f only retains the variability of f over length scales comparable or larger than the filter width D. Though several choices are possible in the present work a Ôtop hatÕ filter: 1 for jxi x0i j < D2 ; 3 0 Gðx; x Þ ¼ D ð2Þ 0 otherwise: is used (Germano, 1992), as it fits naturally into a finite volume formulation.
2. Mathematical model 2.1. LES approach For the purpose of the present computations we use the fence frame of McCluskey et al. (1991), with the x
Fig. 1. Computational domain and relevant parameters.
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Applying the decomposition (1) to the Navier–Stokes equations, for incompressible flows with constant properties, under the hypotheses that filtering and differentiation in space commute, gives rise to: oui o ui o p=q o uj þ ¼ ð rij þ sij Þ oxi oxj ot oxj
ð3Þ
ouj ¼0 oxj
ð4Þ
In Eq. (3), rij ¼ 2msij is the viscous stress tensor, while sij ¼ ui uj ui uj represents the effect of the SGS motion on the resolved motion and is known as the SGS stress. The SGS stress must be modeled before the filtered equations can be solved. The continuity equation (4) remains unchanged on application of the filter, due to its linear nature. 2.2. Sub-grid scale model The model for the SGS stress used in the present work is the dynamic model of Germano et al. (1991) implemented using the approximate localization procedure of Piomelli and Liu (1995) together with a modification proposed by the authors. This model represents the SGS stresses as the product of an SGS viscosity mSGS times the resolved part of the strain tensor. The SGS viscosity is evaluated as the product of the filter length D times an appropriate velocity scale. The appropriate choice for this velocity scale is: Dksk. Thus, for the anisotropic part of the SGS stresses the following expression is found: saij ¼ 2ðCDÞ2 ksksaij
ð5Þ
where the model parameter C needs to be estimated. In the dynamic model this is achieved by applying a second filtering operation, denoted by ~, to Eq. (3). In the test filtered equation the SGS stresses are: Tij ¼ ug uj ui ~ i uj ~
ð6Þ
Comparing the definition of Tij with s~ij we obtain uj ug uj u~i ~ Lij ¼ Tij s~ij ¼ i
ð7Þ
Eq. (7), known as GermanoÕs identity (Germano et al., 1991), involves only known, i.e. resolved, quantities. In order for Eq. (7) to yield the needed expression for C we need to assume some form of relationship between the model constant values C and Cð~Þ at the grid- and testfilter levels. Based on the hypothesis that the cutoff length falls inside the inertial subrange, the approximation conventionally used is: C2 ¼ C2 ð~Þ
ð8Þ
However, such a subrange is not always guaranteed to occur in low-Reynolds number or wall-bounded flows. In particular it is expected that the largest deviations
from universality of the SGS motions will occur in the regions of weakest resolved strain. As a consequence the two values of the model parameter at two different filter levels are likely to differ. In order to account for this it is therefore proposed that: ! 2 2 ð9Þ C ð~Þ ¼ C 1 þ pffiffiffi 2 2 2D k~skk~sa k2 The parameter has the dimensions of dissipation and, if the flow has only one length scale ‘ and one velocity scale v, the following estimate can be used: v3 ‘ though a more refined estimate is probably desirable. With v ¼ U0 and ‘ ¼ hf for the present flow we obtain ¼ 2:02 104 m2 s3 . The reasoning leading to Eq. (9) is that the scale invariance of C can only be invoked if the cutoff falls inside an inertial subrange. When this happens the dissipation provided by the model should represent the entire dissipation present in the flow. On the other hand in the high Reynolds number limit the dissipation is only determined by v and ‘ (with the proviso that the turbulence can be characterised by single length and velocity scales), so that the ratio of to D 2 k~sk3 is a measure of how far the flow is from scale preserving conditions. Eq. (9) represents a first order expansion of the scaledependent expressions for C such as the one proposed by Porte-Agel et al. (2000) using a single length and velocity scale. Making use of Eqs. (7) and (9) yields, on contraction of both sides with the tensor ~s: pffiffiffi 2 ½2 2ðC2 DÞ ksksaij ~ ~saij Laij~saij 2 pffiffiffi 2 C ¼ ð10Þ 2 þ 2 2D k~skk~sa k
where C2 is a provisional value for the field C2 , e.g. its value at the previous time step (Piomelli and Liu, 1995). The dependence embodied in Eq. (9) yields a simple expression for C2 and its evaluation requires only slight modifications of the approximate localization procedure of Piomelli and Liu (1995). Furthermore the formulation is well conditioned and avoids the spiky and irregular behavior exhibited by some other implementations of the dynamic model. It is also to be noted that when the resolved strain tends to zero, C2 also tends to zero, whilst C2 ð~Þ remains bounded. This situation is encountered in the computation of the free stream of external flows. Expression (10) yields smooth C2 fields with no need for averaging. The maxima of C2 thus evaluated are of the same order of magnitude as LillyÕs estimate for the Smagorinsky model constant Lilly (1967), and are located in zones of strong resolved shear, where in low-Reynolds number flow some dissipation is likely to occur at grid filter level. Very small values are
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found in regions where the flow is smooth. The use of Eq. (9) in Eq. (7) amounts to adding a significant diagonal contribution to the integral operator in Eq. (7) thereby improving its conditioning, hence the smooth behavior of C when evaluated according to (10). Eq. (7) provides three independent equations for C2 that cannot be simultaneously satisfied. Contraction of both sides of saij amounts to solving for C2 in a least Eq. (7) with ~ square sense. However, Eq. (10) does not prevent the appearance of negative values of the model parameter, and these values have to be set to zero, in order to prevent instability. Negative values of the SGS viscosity are found on roughly 12% of the grid points at each time step and on these points the SGS viscosity mSGS is set to zero. In the implementation of the model test-filtering is performed in all space directions, and no averaging is performed of the computed model parameter field. The ratio D~=D is set to 2 and the filter width is determined from D ¼ ðDx Dy Dz Þ1=3 . 2.3. Numerical procedure and computational parameters All the computations have been performed with the in house code BOFFIN (Jones, 1991). The code has previously been employed by the authors in the LES of turbulent flow past a backward facing step at Reh ¼ 5100 (di Mare and Jones, 2001), with results in satisfactory agreement with the DNS of Le et al. (1997) and the LES results of Akselvoll and Moin (1997). The code has also been used in the LES of a jet in cross flow (Jones and Wille, 1996a,b), and of vortex shedding past a square cylinder (di Mare and Jones, 1998). The code implements an implicit finite-volume incompressible flow solver using a co-located variable storage arrangement. Because of the co-located arrangement fourth
609
order pressure smoothing, derived from that proposed by Rhie and Chow (1983), is applied in order to prevent spurious oscillations of the velocity field. Time advancement is performed via an implicit Gear method for all transport terms. The overall procedure is second order accurate in both space and time. The time step is chosen by requiring that the maximum Courant number lies between 0.1 and 0.3. This requirement is enforced for reasons of accuracy (Choi and Moin, 1994). The code is parallel and uses the message passing interface MPI-1.2. Computations have been conducted on the Cray T3E-1200 at CSAR, Manchester, using 16, 32, 128 and 512 processors. More details on the numerical algorithm and its implementation are available in di Mare and Jones (2001), Jones and Wille (1996a) and di Mare and Jones (1998) and references cited therein. Several combinations of grids and boundary conditions have been tried. The domain and grid sizes for all the test cases are shown in Table 1, together with grid spacings in wall-units, based on the maximum friction coefficient from the experiments (0.002). The computations with free-slip upper boundary, A and B, give attachment lengths XA ¼ 15:6 hf and 17.3 hf , respectively, compared with the experimental value of XA ¼ 21 hf , the better value coming from the more resolved computation. The computations with entraining upper boundary, C, D, E and F give XA ¼ 17:6 hf , 24 hf , 18.5 hf and 21.8 hf , respectively, as shown in Table 1. The domains used for runs C and D are very narrow and are most to affect the computed results. For this reason runs D and E have been performed. These two runs have the same grid, twice as wide as runs C and D, better resolved in the span-wise and stream-wise directions, but with the same resolution in the wall-normal direction. In run E the dissipation parameter has been evaluated as ¼ 0:001v3 =l while in run D is evaluated as ¼ v3 =l.
Table 1 Mesh parameters max Dxþ i
min Dxþ i
Expansion ratio
XA
4.90 4.40
1.28 1.16 1
15.6
0.52
1 1.04 1
17.3
67.1 135.0 34.1
1.45
1 1.05 1
17.6
59.1 29.1 3.39
33.8 136.0 34.1
1.41
1 1.04 1
24.0
41.4 29.1 6.19
24.6 136.0 10.4
1.41
1 1.04 1
18.5/21.8
Case
Direction
Points
Li =hf
A
x y z
204 60 80
50.2 7.69 10.0
87.3 59.0 20.7
x y z
282 148 84
40.3 13.8 6.38
23.2 72.2 13.6
x y z
204 116 21
58.5 29.1 3.39
x y z
404 116 21
x y z
388 116 116
B
C
D
E/F
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The differences in the computed results from tests E and F show that the model is indeed active also at high resolutions and low Reynolds numbers. Time averages are computed as integrals in time. Statistics have been accumulated for at least 300 hf =U0 for all test cases. The inflow boundary conditions are represented with a uniform velocity U0 . No perturbations are imposed. This choice is justified by the very low level ( 0.3%) of free-stream turbulence found in the target experiments of Hardman (1998). The inflow conditions are applied 10 hf upstream of the fence. No numerical oscillations are observed upstream of the fence. The outflow boundary is treated as a convective outflow (Pauley et al., 1990), i.e. oui oui ¼ uc ot ox
ð11Þ
where the convection velocity uc is the local time-average velocity vector evaluated at the outflow plane. The upper boundary is treated as a free-slip, non-porous wall for test cases A and B, as an entraining boundary for test cases C and D. The entraining boundary is represented with an homogeneous Neumann boundary for all velocity components and as a Dirichlet boundary condition for the pressure. The pressure field is the one corresponding to the potential flow in the free stream, i.e. is assumed uniform: oui ¼0 on
ð12Þ
p¼0
ð13Þ
Due to the large effect of the blocking ratio on the experimental results of Hardman (1998), a very high computational domain is used. The side boundaries are periodic. No-slip conditions are applied at wall boundaries and no wall-model is used. The lower boundary of the inflow section is also treated as a non-porous, freeslip wall.
Fig. 2. Instantaneous contours of SGS dissipation (a) and entrophy (b) on the centre-plane. Only part of the computational domain is shown.
of the SGS dissipation can be seen to occur across the shear layer bounding the separated flow and again across the boundary layer developing downstream of attachment. No significant dissipation occurs near the walls, nor, of course, in the undisturbed flow. The average behaviour of the model is shown in Figs. 3 and 4. More precisely Fig. 3 shows profiles of the time averaged model parameter C2 at several locations downstream of the fence. The highest values of C2 occur close to the obstacle, and decay quite rapidly at downstream locations. It is to be noted that the average value of C2 is always much smaller than the commonly used values of the Smagorinsky parameter, even though values similar to those reported are indeed attained instantaneously. The computed average of the SGS uvReynolds stress is shown in Fig. 4. Consistently with the behaviour of C2 , the SGS Reynolds stresses are highest close to the fence and then decay rapidly further downstream. Figs. 3 and 4 also show the effect of the dissipation parameter on the model parameter and SGS stresses. More precisely, in both runs D and F is evaluated as ¼ v3 =‘ and the model parameter C and the SGS stresses from the two runs are comparable, while in run E ¼ 0:001v3 =‘ and the C has values comparable with LillyÕs estimate throughout the boundary layer, except near the wall. Interestingly the SGS stresses from run E do not differ from those obtained from runs D and F as dramatically as the model parameters do. 3.2. Instantaneous flow field
3. Results and discussion 3.1. Model and SGS stresses Results concerning the time-dependent behaviour of the SGS model, Eq. (10) are shown in Fig. 2. Fig. 2(a) shows the instantaneous contours of SGS dissipation in the midplane. The high values of SGS are clustered across the shear layer bounding the separated flow region and, to a lesser extent, across the developing boundary layer downstream of attachment. The dissipation contours follow very closely the pattern of the resolved entrophy contours, shown in Fig. 2(b), which is consistent with the discussion in Section 2.2. Almost all
More results concerning the time-dependent behaviour of the flow field for this configuration are shown in Fig. 5. Fig. 5(a) presents instantaneous stream-wise velocity contours near the splitter plate at y=hf ¼ 0:09. The different structure of the flow field beneath the separated flow and beneath the developing boundary layer downstream of attachment are clearly visible. No well defined ‘‘streaks’’ are found in the flow even far downstream of attachment, indicating that the flow structure has not evolved to that of an attached boundary layer even by two attachment lengths downstream of the fence. No such structure is discernible in the separated flow region either. Some interesting observation can be
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Fig. 3. SGS model parameter at several locations downstream of the fence. - - -: run D, : run E, ––: run F.
Fig. 4. SGS uv Reynolds stress at several locations downstream of the fence. - - -: run D, : run E, ––: run F.
Fig. 5. Stream-wise (a) and span-wise (b) velocity contours near the splitter plate. Dark contours: positive values, light contours negative values, black contours zero.
made with the respect to the behaviour of the span-wise velocity component close to the wall. Beneath the separated region and close to the wall, the average value of the span-wise velocity must be positive, yet no dominant span-wise flow direction can be deduced from Fig. 5(b). The fact that the instantaneous span-wise velocity component bears very little resemblance to its mean pattern suggests that the span-wise flow in the configuration of interest is highly unsteady; consistent with the observations of Hardman (1998).
3.3. Flow statistics Fig. 6 shows the pressure and friction coefficients for test cases D, E and F compared to the experimental results (Hardman, 1998). The agreement is quite good in the separated region, especially for x=XA < 0:7. Some discrepancies can be observed in the recovery region. The friction coefficient in the z direction, Cfz also appears too low. The agreement in the separated flow region appears to be better for runs E and F, and it is
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Fig. 6. Pressure and friction coefficients. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
worse for run E. Further downstream all runs exhibit too low a Cfz : this behaviour is still being investigated. The average profiles of U =U0 and W =W0 are shown in Figs. 7 and 8, together with the measurements by Hardman (1998). The agreement for the u component is very good at all stations, while the w profiles are reasonably well reproduced, though some features are missing in the computational results, especially near the
wall, consistently with the results for the friction coefficients. Fig. 9 shows a comparison of the LES results from test cases D, E and F with SimpsonÕs law (Simpson, 1983), for separated flows far from attachment points.
y
y U
ð14Þ ¼A log 1 1 UN N N
Fig. 7. Average u-component velocity profiles at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
Fig. 8. Average w-component velocity profiles at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
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Fig. 9. Average u-component velocity profiles at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
where UN is the maximum negative velocity and N is the distance from the wall where it is achieved. A is a universal parameter with a commonly accepted value of A ¼ 3. The LES data exhibit a logarithmic region between y=N ¼ 0:02 and 0.2 that agrees with Simpson law for distances from the fence between 0:25XA and 0:75XA , while the profiles at 0:125XA and 0:875XA , as expected, do not follow the logarithmic law. The discrepancies between runs E and F show that the effect of the model in early stages of separation is to change the shape of the separated region and hence the position of the maximum negative velocity. Figs. 10–13 show turbulence intensities and resolved Reynolds stress profiles at several stations downstream of the obstacle. Here again agreement with the experimental results for run D is reasonable in the first
half of the separated region and worsens further downstream. The under prediction of the Reynolds stress is consistent with the under prediction of Cp and Cfx in the recovery region. Here the narrow domain precludes the representation of large structures that contribute significantly to the Reynolds stresses. This effect is not important upstream of attachment because there are no such large structures there, as the boundary layer at separation is laminar and the eddy structures have not grown to such an extent as to saturate the computational domain. This goes someway to explain the previous statement regarding Cp and Cfx . That this diagnosis is correct is supported by the better agreement between experiments and LES data for the Reynolds stresses and turbulence intensities for runs E and F.
Fig. 10. u-component turbulence intensity at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
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Fig. 11. v-component turbulence intensity at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
Fig. 12. w-component turbulence intensity at several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
Fig. 13. uv Reynolds stress several locations downstream of the fence. - - -: run D, : run E, ––: run F, j: reference data (Hardman, 1998).
4. Conclusions Some results from the LES of the spanwise invariant flow past a swept fence have been presented. The in-
stantaneous and time-average behaviour of the flow field is correctly reproduced by the method at a reasonably low computational expense. The paper has proposed a variant of the dynamic SGS model of Germano et al.
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(1991) which appears to be well suited to comparatively low-Reynolds number computations. The model is based on the possibility of allowing different values of the Smagorinsky constant at different filter levels. The model parameter is numerically well behaved and does not need averaging of the computed fields, thus making its extension to complex geometries straightforward and computationally attractive. Consistent with the experiments, the computations have been found to be very sensitive to spurious blocking effects, so that the best results have been obtained with a very high domain and entraining boundaries. Future work in the flow configuration will involve a better estimate for the dissipation scale in the proposed sub-grid model. Further computations are currently being run with a wider domain in the spanwise direction in order to remove the confinement effect shown in the results currently available.
Acknowledgements The authors are grateful to Dr. P.E. Hancock for providing experimental data and for helpful discussions. The authors also acknowledge support from EPSRC under grant GR/N17683.
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