2.1. Stretched vortex SGS model

In LES, the velocity components $u_{i}$ and pressure $p$ can be decomposed as a resolved part and an unresolved part: $u_{i}=\tilde{u_{i}}+u_{i}^{\prime }$ , $p=\tilde{p}+p^{\prime }$ . After applying a filter with scale  ${\it\Delta}_{c}$ , the filtered Navier–Stokes (N–S) equations can be written as

with the subgrid stress (SGS) tensor  $\unicode[STIX]{x1D61B}_{ij}\equiv \widetilde{u_{i}u_{j}}-\widetilde{u_{i}}\widetilde{u_{j}}$ . The subscript $i$ or $j$ presently denotes three components $x$ , $y$ and $z$ , corresponding to the streamwise, spanwise and wall-normal directions, respectively. The three velocity components are also written as $u$ , $v$ and $w$ .

The stretched-vortex SGS model owes its genesis to the stretched-spiral vortex model by Lundgren (Reference Lundgren1982). The model assumes that in each computational cell, the subgrid motion is dominated by a vortex with direction $\boldsymbol{e}^{v}$ , modelled by a delta-function probability density function. The SGS tensor is modelled as Chung & Pullin (Reference Chung and Pullin2009):

with $k_{c}={\rm\pi}/{\it\Delta}_{c}$ the cutoff wavenumber, $k$ the wavenumber, and $E(k)$ the energy spectrum. In the integration above, the Lundgren vortex model is used, with $\mathscr{K}_{0}^{\prime }=\mathscr{K}_{0}{\it\epsilon}^{2/3}{\it\lambda}_{{\it\nu}}^{2/3}$ , ${\it\Gamma}$ is an incomplete Gamma function with ${\it\kappa}_{c}=k_{c}{\it\lambda}_{{\it\nu}}$ , ${\it\lambda}_{{\it\nu}}=(2{\it\nu}/(3|\tilde{a}|))^{1/2}$ , $\tilde{a}=e_{i}^{v}e_{j}^{v}\widetilde{\unicode[STIX]{x1D61A}}_{ij}$ is the stretching along the subgrid vortex and $\widetilde{\unicode[STIX]{x1D61A}}_{ij}=(\partial \widetilde{u}_{i}/\partial x_{j}+\partial \widetilde{u}_{j}/\partial x_{i})/2$ is the resolved strain-rate tensor.

By using a matching procedure, the constant $\mathscr{K}_{0}^{\prime }$ can be found as $\mathscr{K}_{0}^{\prime }=\langle F_{2}\rangle /\langle Q({\it\kappa}_{c},d)\rangle$ . Here $\langle ~\rangle$ denotes the local average and is computed technically from a set of 26 neighbouring points, and $F_{2}$ is the local second-order velocity structure function calculated from the resolved velocities. The quantity

is calculated using an efficient asymptotic approximation with $d=r/{\it\Delta}_{c}$ and $r$ is the distance from neighbour point to the vortex axis (Chung & Pullin Reference Chung and Pullin2009).

The SGS model discussed above is utilized in the main body of the flow. For modelling near-wall flow, Chung & Pullin (Reference Chung and Pullin2009) developed an extended SGS model based on the idea that the convection of both the passive scalars and the axial velocity inside cylindrical SGS vortices can be analysed within a two-dimensional, three-component flow framework. This leads to an additional term added to the original SGS model

with $K_{s}={\it\gamma}{\it\Delta}_{c}K^{1/2}/2$ and ${\it\gamma}$ a ‘mixing’ constant. This SGS model was used in the virtual-wall model to obtain the wall-normal flux of wall-parallel momentum in the near wall region.

2.2. Wall model

Chung & Pullin (Reference Chung and Pullin2009) argue that the dominant parameter for near wall-modelling is $u_{{\it\tau}}$ . In order to calculate $u_{{\it\tau}}$ without resolving the fine scales near the wall, they introduce wall-normal averaging of the wall-parallel, streamwise (outer flow) momentum equation combined with local inner scaling used to treat the time-dependence of the filtered, wall-parallel velocity. This latter assumption requires no specific form for the wall-normal variation of this velocity. This results in an ODE describing the local wall-normal velocity gradient ${\it\eta}_{0}=\partial \tilde{u} /\partial z|_{w}$ or equivalently $u_{{\it\tau}}^{2}\equiv {\it\nu}\,{\it\eta}_{0}$ , at each wall point with coefficients and source terms that, with a few-point approximation to certain wall-normal integrals, can be determined dynamically from the outer LES at the first few grid points away from the wall. Combined with a log-based description of the slip velocity at a virtual wall at a distance $h_{0}$ (to be defined later) from the actual wall that is a small fraction of the first wall-normal grid location, this provides closure.

Following this approach, we begin with the Navier–Stokes equations and apply two filters:

where (2.5) defines the wall-parallel filtered ( $xy$ -plane) quantity $\tilde{{\it\phi}}$ and (2.6) defines a wall-normal ( $z$ ) or top-hat filter $\langle {\it\phi}\rangle$ . For the wall-parallel velocity components these are presently $\tilde{u}$ and $\tilde{v}$ and $\langle u\rangle$ and $\langle v\rangle$ , respectively. We remark that presently, the wall-parallel filtering is strictly formal.

After applying the wall-parallel filter to both the streamwise and spanwise momentum equations, neglecting lateral diffusion, we obtain

3.1. Numerical method

The LES equations, including the resolved-scale Navier–Stokes equations and the ODE for ${\it\eta}_{0}$ , are integrated using the three-stage low-storage Runge–Kutta method of Spalart, Moser & Rogers (Reference Spalart, Moser and Rogers1991) for temporal evolution. For each stage, the fractional-step method (Perot Reference Perot1993) is implemented for the temporal-discretized Navier–Stokes equation which results in a modified Helmholtz equation for the velocity arising from an implicit treatment of the viscous terms, a Poisson equation for pressure and finally a correction step. For spatial discretization, we employ a staggered grid in the streamwise/wall-normal plane in which the streamwise (respectively wall-normal) velocity components are stored at the $x$ (respectively $z$ ) faces of the computational cells and cell-centred storage in the spanwise direction. In obtaining Helmholtz equations, both convective and nonlinear terms are evaluated using a fourth-order finite difference scheme in the $x$ and $z$ -directions, and a pseudo-spectral representation for the $y$ -direction. A skew-symmetric form for convective terms is employed to improve energy conservation and reduce aliasing errors. The Poisson equation for pressure is reduced to a series of one-dimensional equations in the $z$ -direction by applying a spectral method in the $y$ -direction and a fast-cosine transform in the $x$ -direction. Finally, the Helmholtz equation and the one-dimensional Poisson equations are solved by the septa-diagonal matrix solver provided by LAPACK. The above methods, originally implemented by Inoue & Pullin (Reference Inoue and Pullin2011), have been validated for both DNS and LES of TBL flows.

The present code is designed as an ‘AB code’ framework, with code-A for a prior zero pressure gradient turbulent boundary layer (ZPGTBL) simulation and code-B for the main separation/reattachment LES. The computational domains for both codes are of rectangular parallelepiped shape. Code-A and code-B employ the same numerical method as discussed above, but employ different boundary conditions. For comparison, table 1 lists all the boundary conditions for both code-A and code-B. The same boundary conditions are used in the two codes for the outflow boundary and the virtual wall boundary. The inflow boundary condition for code-A is generated by the recycling method (Lund, Wu & Squires Reference Lund, Wu and Squires1998). In code-A, a Dirichlet pressure boundary condition is used for the top boundary. Since code-B does not have uniform free-stream flow at the upper boundary, the most accurate boundary condition for the top boundary is to specify the normal velocity profile $W_{top}(x,t)$ . This type of boundary condition was used in DNS of separation in a flat-plate TBL by Na & Moin (Reference Na and Moin1998) and Skote & Henningson (Reference Skote and Henningson2002).

Table 1.

Boundary conditions for A and B codes. D, Dirichlet type; R, recycling method (Lund et al. Reference Lund, Wu and Squires1998); C, convective outflow.

3.2. Simulation strategy

Presently, we try to match wall conditions in experiments performed with curved upper walls using LES in rectangular parallelepiped domains. For flow where there exists a separation bubble, care is required in making comparisons between LES and the results of these experiments. Our general strategy is to design boundary conditions for code B that will reproduce flat-wall, streamwise pressure distributions close to those measured in the experiments. This, together with a parametric description that characterizes flows with separation/reattachment is now discussed in some detail.

Unlike the ZPGTBL, which has a uniform free-stream flow and only one parameter (Reynolds number), flow with separation from a flat plate will depict features that depend critically on the top boundary condition. Suppose that the form of the vertical velocity function along the top wall is given by $W_{top}(x/L_{p})$ , where $L_{P}$ is a length scale. Then for a given rectangular parallelepiped computational domain, the remaining flow parameters would be the inflow velocity $U_{\infty }$ , the viscosity ${\it\nu}$ , some measure of the boundary layer thickness ${\it\delta}$ and $L_{p}$ . For diagnosing flow results, two other parameters of interest are the length $L_{B}$ and the height $H_{B}$ of the separation bubble. Using dimensional analysis, we can then write

Generally the functional form of both $F_{1}$ and $F_{2}$ may be expected to depend on the specific form of $W_{top}(x/L_{p})$ .

In order to produce flow containing a separation bubble, the imposition of a streamwise pressure distribution is required. Here $C_{p}$ increases with the imposed APG prior to boundary-layer separation. Inside the separation bubble itself, $C_{p}$ will be determined by the structure the bubble flow, typically remaining approximately constant for the first part of the separation bubble, and then increasing to an apex or maximum within the incipient reattachment region. After reattachment, $C_{p}$ then decreases in a favourable-pressure-gradient (FPG) region to match a desired outflow boundary condition. In order to discuss this in quantitative terms, we summarize our interpretation of some prior work on the general effect of pressure gradient as follows. We consider the smooth development of $C_{p}(x)$ along a streamwise direction, and define two parameters $C_{inc}$ and $C_{dec}$ , the streamwise average of changes in $C_{p}$ during separation and reattachment, respectively, with $C_{inc}=|{\rm\Delta}C_{p}|_{i}/{\rm\Delta}x_{i}^{\prime }$ and $C_{dec}=|{\rm\Delta}C_{p}|_{d}/{\rm\Delta}x_{d}^{\prime }$ . Here the subscript ‘ $i$ ’ denotes the increasing part before separation and ‘ $d$ ’ the decreasing part after reattachment and ${\rm\Delta}x^{\prime }$ is the dimensionless length of the region over which $C_{p}$ changes, generally defined as ${\rm\Delta}x^{\prime }=L_{p}/{\it\delta}_{99}^{in}$ where ${\it\delta}_{99}^{in}$ is the 99 % boundary-layer thickness just before the onset of the imposed APG. In the present simulation the APG corresponds to the first part of $W_{top}(x)$ on the top boundary.

Figure 1.

Sketch configuration of the separation-bubble flow after Perry & Fairlie (Reference Perry and Fairlie1975).  $L_{in}$ , length of ZPG part near inlet; $L_{out}$ , length of ZPG part near outlet; $L_{P}$ length of APG part; $L_{B}$ , length of the bubble; $H_{B}$ , height of the bubble.

Table 2.

Table of parameters for experimental and DNS data. $C_{inc}$ and $C_{dec}$ correspond to the average spanwise increase and decrease in pressure coefficient $C_{p}$ , respectively. All $Re_{{\it\theta}}$ , $L_{p}$ and ${\it\delta}_{99}$ are calculated from data or scaled from figures in reference cited.

a Estimated from data on ${\it\delta}^{\ast }$ . Ratio of ${\it\delta}^{\ast }/{\it\delta}_{99}$ from Chauhan, Monkewitz & Nagib (Reference Chauhan, Monkewitz and Nagib2009).

b From Skote (Reference Skote2001).

c Insufficient information available.

Table 2 lists our estimates of these parameters for some previous work on separation in TBL flow. We find that Perry & Fairlie (Reference Perry and Fairlie1975) has a very mild APG but a strong FPG. Patrick (Reference Patrick1987) has both a strong APG and subsequent FPG, while Lögdberg et al. (Reference Lögdberg, Angele and Alfredsson2008) has a mild APG for separation and weak FPG for reattachment. One aim of the present work is to undertake strong quantitative comparisons between our LES and some of the existing experimental studies of separation and reattachment. We first address the case of Perry & Fairlie (Reference Perry and Fairlie1975) in order to test the present wall model up to separation, and also for a mild APG followed by a strong FPG at low Reynolds number. Subsequently, we address the flow studied by Patrick (Reference Patrick1987) to evaluate the performance of the overall LES, including the present two-dimensional wall model at moderately high Reynolds number.

4.1. Domain size and mesh resolution

Figure 2.

Vertical velocity distribution $W_{top}(x)$ used as a boundary condition on the top-wall of a rectangular parallelepiped domain for the case of Perry and Fairlie. – – – –, $W_{top}=0$ .

Figure 1 provides a sketch for the interpretation of the present LES in relation to the experimental conditions. This illustrates the definitions described previously, including $L_{p}$ , $L_{B}$ , $H_{B}$ and $Re_{{\it\delta}}$ . We identify two distinct parts or portions of the ZPGTBL, before separation and after reattachment, with lengths defined as $L_{in}$ and $L_{out}$ , respectively. These connect to the non-zero pressure gradient part of the flow. These two ZPG parts are used to accommodate the flow to ensure that the inflow and outflow boundary conditions have minimal impact on the flow within the non-zero pressure gradient region. In figure 1 the experimental configuration is sketched as a dual-roof top wall which generates first the APG followed by the FPG. The present LES employs instead a parallelepiped box. The present strategy is then to match the given experimental $C_{p}(x)$ data, to be discussed subsequently, on the bottom wall by tuning $W_{top}(x)$ . This is done by trial and error using a rough, simple potential flow model using sinks, sources and vortices as a guide. The final $W_{top}(x/{\it\delta}_{0})$ used for the Perry & Fairlie (Reference Perry and Fairlie1975) experiment is plotted in figure 2.

We use a nominal boundary-layer thickness ${\it\delta}_{0}$ as a length scale and fix the 99 % boundary-layer thickness at the beginning of the APG region as $1.2{\it\delta}_{0}$ . The domain length in the streamwise direction was set to be $216{\it\delta}_{0}$ , with $W_{top}(x/L_{p})$ enforced in a length $L_{p}=170{\it\delta}_{0}$ . This numerical test section joins an upstream ZPG part with $L_{in}=10{\it\delta}_{0}$ , which is for the development of the inflow, and is connected to a second, downstream ZPG part with $L_{out}=36{\it\delta}_{0}$ . The ZPG parts are designed to ensure that the influence of vertical velocity enforcement does not reach the inflow and outflow boundaries in the streamwise direction. As a check, we plot isolines of the mean (in the sense of spanwise and time-averaged) vertical velocity in the $(x,z)$ plane of the simulation domain in figure 3 for a typical LES. It can be seen that the isolines of $w=\pm 0.01$ , which correspond to approximately 1 % of the maximum vertical velocity, do not extend to either the inflow or outflow boundaries.

Figure 3.

Isolines of spanwise mean vertical velocity $W$ . $W=0.01$ describes the upper limit of the desired vertical velocity for an acceptable upstream influence of the APG.

Figure 4.

Isolines of mean streamwise velocity $U$  (– – – –) and streamlines of mean velocity (—— with arrows). Line — ⋅ — is the isoline of $U=0.4595$ , a visualization of the saddle point.

The domain size in the wall-normal direction should be restricted by a fully-potential flow top boundary criterion. We ensure that the saddle point of the $\tilde{u} (x,y)$ field above the separation bubble is located within the computational domain. A similar criterion was used by Na & Moin (Reference Na and Moin1998). Figure 4, in which the dashed lines are the isolines of mean streamwise velocity while the solid lines are the streamlines of the mean velocity, illustrates this criterion. The domain in the $z$ -direction is presently $24{\it\delta}_{0}$ . This is also where the vertical velocity boundary condition $W_{top}(x)$ , shown in figure 2, is imposed.

There is no straightforward criterion available for the domain size in the spanwise direction. Ideally the spanwise domain should have sufficient space for the development of the largest turbulent structures. However, as wall-modelled LES cannot precisely capture these largest structures, the present strategy is to compare the averaged properties from results predicted with a different spanwise domain. We start with a spanwise domain $12{\it\delta}_{0}$ , which is comparable to the domain size of $9.8{\it\delta}_{0}$ in Na & Moin (Reference Na and Moin1998).

Presently, several LES cases are implemented in order to assess the influence of domain-size and resolution; see table 3. Three coarse mesh cases test domain size, with $C_{0}$ the baseline domain discussed above, $C_{1}$ a smaller domain in the wall-normal direction $L_{z}=18{\it\delta}_{0}$ and $C_{2}$ is the doubled spanwise domain. Case $F_{1}$ is a fine mesh case for assessing mesh resolution. Note that for all cases with $L_{z}=18{\it\delta}_{0}$ , the vertical velocity at $z=18{\it\delta}_{0}$ obtained from case $C_{0}$ is imposed on the top boundary. The Reynolds number is set to $Re_{{\it\theta}}^{in}=2000$ with superscript ‘ $in$ ’ denoting just upstream of the region where the vertical velocity boundary condition is imposed. This is implemented by adjusting the inflow boundary-layer thickness in code-A. For most LES, model I was used for calculation of ${\it\theta}_{0}$ . One case was performed using model II.

Figure 5.

Wall pressure coefficient $C_{p}$ : ▫, experiment (Perry & Fairlie Reference Perry and Fairlie1975), – – – –, case $C0$ ; — ⋅ —, case $C1$ ; — ⋅ ⋅ —, case $C2$ ; $\cdots \cdots$ , case $C3$ ; ——, case  $F1$ .

Figure 6.

Time evolution of maximum height of the backflow region, $h_{B}$ . First $3T_{t}$ is the start-up transient followed by $5T_{t}$ for stabilizing the flow and then a final $10T_{t}$ for diagnostics and output. $T_{t}$ is a typical particle transit time across the domain.

Figure 7.

Boundary-layer thickness parameters up to separation. Symbols for experiment (Perry & Fairlie Reference Perry and Fairlie1975), lines for LES results, case $F1$ . ▫, $Re_{{\it\theta}}$ , ○, $H$ . ——, $Re_{{\it\theta}}$ , - - - -, $H$ .

Table 3.

LES performed for Perry & Fairlie (Reference Perry and Fairlie1975) flow: three coarse-mesh cases with different spanwise and wall-normal domain dimensions, and one finer-mesh case, $F1$ . $Re_{{\it\theta}}^{in}=2000$ is held constant. $C3$ is an extension of $C1$ with using model II.

4.2. Wall streamwise pressure distribution

As shown in figure 5, $C_{p}$ distributions from the various LES are similar and they all match reasonably with the experimental measurements of Perry & Fairlie (Reference Perry and Fairlie1975). Figure 5 also includes a close-up of $C_{p}$ in the separated region. We find that the differences between cases are within model error, and that the fine mesh case $F1$ shows somewhat better agreement with experimental data than the coarse mesh cases. We note also that the LES using model I for ${\it\theta}_{0}$ is very close to that using model II at a similar resolution.

The initial conditions used for both code A and code B are zero-pressure gradient results from previous LES at the same Reynolds number. The vertical velocity boundary condition on the top boundary is applied instantaneously from $t=0$ . Figure 6 shows the time evolution of the maximum height of the backflow region $h_{B}$ . This rapidly increases from zero, reaching a maximum at about three times a characteristic flow time $3T_{t}$ where $T_{t}$ refers to the time required for a typical free-stream particle to transit the full flow domain. At about $t=8T_{t}$ , the flow is considered to reach statistical steady state, and simple average diagnostics are implemented in the next $10T_{t}$ period.

4.3. Boundary-layer thickness and skin-friction distribution

Because Perry & Fairlie (Reference Perry and Fairlie1975) recorded boundary-layer flow data prior to flow separation, we focus first on the APG of the flow. Figure 7 shows the streamwise variation of both $Re_{{\it\theta}}$ and the shape factor $H\equiv {\it\delta}^{\ast }/{\it\theta}$ , respectively versus $x/{\it\delta}_{0}$ prior to separation. It can be seen that our LES generally matches the $Re_{{\it\theta}}$ number development in this region.

Figure 8.

Skin friction coefficient $C_{f}$ . Symbols for experiments by Perry & Fairlie (Reference Perry and Fairlie1975), lines for present LES. ▫, $C_{f}$ ; ○, separation point; ▵, reattachment point; – – – –, case $C0$ ; — ⋅ —, case $C1$ ; — ⋅ ⋅ —, case $C2$ ; $\cdots \cdots$ , case $C3$ ; ——, case $F1$ . Horizontal dashed line indicates $C_{f}=0$ .

The skin friction $C_{f}$ is defined presently as

where the overline refers to a time-spanwise average, and we note that, by definition, $C_{f}\geqslant 0$ always. This is perhaps the most important flowfield parameter in our wall-modelled LES as we directly solve for ${\it\eta}_{0}$ from the ODE (2.14). Figure 8 plots $C_{f}$ versus $x/{\it\delta}_{0}$ . It is seen that our LES can reasonably capture the $C_{f}$ development in the streamwise development prior to separation where measurements were recorded. This means the whole APG TBL development before separation is accurately represented by the LES. This result is consistent with the APG TBL simulation performed by Inoue et al. (Reference Inoue, Pullin, Harun and Marusic2013), which uses Dirichlet streamwise velocity and pressure boundary conditions. Note that in Inoue et al. (Reference Inoue, Pullin, Harun and Marusic2013), a one-dimensional wall model is used and only a small APG region is implemented. Again it can be seen that the $C_{f}(x)$ variation using model I for ${\it\theta}_{0}(x,y,t)$ is very close to that obtained using model II. This is evidence that the overall LES is relatively insensitive to this choice. Hence subsequently, all LES discussed presently utilize the simpler model I.

Two points plotted in figure 8 are the separation point and reattachment point obtained from experiment from mean-flow streamlines. In the present simulation, the two-dimensional wall model enables calculation of $C_{f}$ up to and through separation. For three-dimensional flow, even when statistically stationary in some sense, a ‘separation point’ is not generally defined. But since the present flow is driven by boundary conditions that have no variation in the $y$ or spanwise direction, it is expected to be statically stationary and two-dimensional in the time-wise mean. Hence presently, the separation and reattachment points in $x$ are defined as those points where the time-spanwise averaged $x$ -shear stress $\overline{{\it\tau}_{w,x}}$ , as calculated from the ODE, changes sign. As shown in figure 8, this measure of the separation and reattachment points agrees well with the experimental results.

The separation/reattachment flow is highly unsteady and far from two dimensional. This is illustrated in figure 9 which shows an $x{-}y$ plane plot, on the plane $z=0$ , of the instantaneous (at a particular time) streamwise component of wall shear stress ${\it\tau}_{w,x}$ . The whole-domain plot on the figure 9(a) shows the spatially non-uniform character of a typical instantaneous distribution of ${\it\tau}_{w,x}$ . From the close-up plot, we can see that separation is not well defined in an instantaneous image. In the region surrounding the mean separation line as defined previously, at about $x/{\it\delta}_{0}=112$ , the instantaneous field of ${\it\tau}_{w,x}$ is highly irregular, showing patches and areas of both positive and negative sign over a region of extent many ${\it\delta}_{0}$ in the streamwise direction. This feature is perhaps associated with the moderate APG for this case. In contrast, further downstream, the ${\it\tau}_{w,x}$ field appears to be more ordered in the reattachment region where the strong FPG enables recovery of the skin friction to quite strong positivity.

4.4. Bubble size and velocity profile

To compare data that characterizes the extent of the separation bubble, we compare both $L_{B}/{\it\delta}_{99}^{in}$ and $H_{B}/{\it\delta}_{99}^{in}$ in table 4. Note that here $L_{B}/{\it\delta}_{99}^{in}$ is defined by the separation point and reattachment point from $C_{f}=0$ . $H_{B}/{\it\delta}_{99}^{in}$ is evaluated from the maximum height of the enveloping line of the separation bubble. Both the bubble length and height in all simulations are close to the experimental data within a few per cent.

Figure 9.

Contour plot of the instantaneous ${\it\tau}_{w,x}(x,y)$ . Case  $F1$ of table 3. (a) The whole domain. (b) Close-up near separation. Both figures use the key shown.

Table 4.

Comparison of bubble size with experimental data from Perry & Fairlie (Reference Perry and Fairlie1975): length of the bubble $L_{B}$ and height of the bubble $H_{B}$ . Length scale ${\it\delta}_{99}^{in}$ set to be ${\it\delta}_{99}$ before APG part of flow.

Figure 10.

Comparison of velocity profiles. ▪, Experimental data Perry & Fairlie (Reference Perry and Fairlie1975); ——, LES case $C1$ ; – – – –, LES case $F1$ .

Figure 10 shows the mean streamwise velocity profile at different locations. These match those of the experiment, with two upstream of the bubble ( $x/{\it\delta}_{0}=102.6,110.6$ ), four points inside the bubble ( $x/{\it\delta}_{0}=121.6,131.0,139.4,150.0$ ) and one final point downstream the bubble ( $x/{\it\delta}_{0}=160.0$ ). We plot velocity profiles for two cases, a solid line for $C1$ and dashed line for $F1$ . It can be found that, in the region upstream of the bubble, the velocity profiles obtained from LES are in reasonably good agreement with experiments.

Skote & Henningson (Reference Skote and Henningson2002) propose a model for mean velocity scaling in the separation region. Prior to separation, their mean velocity is given by

while the velocity profile inside the separation region is

where

Figure 11.

Comparison of velocity profiles: $\cdots \cdots$ , log line $U^{+}=(1/0.378)\ln z^{+}+4.08$  (Inoue & Pullin Reference Inoue and Pullin2011); — ⋅ —, (4.2); – – – –, (4.3); —— with symbols, LES results. ○, $x/{\it\delta}_{0}=102.6$ ; ▫, $x/{\it\delta}=121.6$ .

The present LES is compared with these scalings in figure 11 where we use the same parameters as suggested in  Skote & Henningson (Reference Skote and Henningson2002): ${\it\kappa}=0.41$ , $B=1.5$ and $C=-7$ . In determining the LES profiles, we presently use $u_{{\it\tau}}$ defined in (4.4). This is strictly for the present comparison and is not the same as the wall-model definition of $u_{{\it\tau}}$ given by the second of (2.9). We compare two velocity profiles at $x/{\it\delta}_{0}=102.6$ and $121.6$ , which are the first and third velocity profiles in figure 10. The velocity profile at $x/{\it\delta}_{0}=102.6$ , which is in the APG region, agrees quite well with (4.2), but is different from the classical log law. The velocity profile within the separated region also matches (4.3) reasonably well up to $z^{+}\approx 350$ , implying that the Skote and Henningson theory is valid for weakly separated flow.

Figure 12.

Velocity in backflow region. (a) Isolines of mean streamwise velocity of backflow, $U<0$ (contour of  $U$ ). Velocity probed at $P1$ , $P2$ , $P3$ and $P4$ (vertical dash lines) are extracted for (b). (b) Scaled velocity. ——, Scaling equation (4.5), Simpson (Reference Simpson1983); – – – – with symbols, LES results. ▵, $P1$ ; ▫, $P2$ ; ○, $P3$ ; ♢, $P4$ (comparison of scaled  $U$ ).

For the region downstream of separation, Simpson (Reference Simpson1983) suggests a scaling formula to describe the velocity profile of the main backflow region. In this scaling, the maximum negative velocity $U_{N}$ and its distance from the wall $N$ are two basic parameters. Simpson’s equation, which covers the region of $0.02<z/N<1.0$ with $z$ the wall-normal coordinate, is given by

with $A=0.3$ . To compare with this scaling formula, contours of the backflow velocity are first plotted in figure 12(a). This shows that the present LES can capture the maximum negative velocity at about $135<x/{\it\delta}_{0}<145$ . The streamwise velocity at four positions are extracted and plotted in figure 12(b), with $x/{\it\delta}_{0}=133.5(P1)$ , $136.5$ , $140$ and $145$ . We find that at position $P2$ , which is a good example from the LES owing to obvious maximum backflow velocity and sufficient mesh points for $z<N$ , the LES results agree quite well with the formula. For $P1$ , there is interaction of different isolines. This complexity raises the backflow velocity to somewhat above the theory. The velocity profiles at position $P3$ also agrees well with the scaling formula. Comparison of $P2$ and $P3$ shows that only the first two near-wall points deviate from the formula, with fair agreement for other points. Except for $P4$ , where there are only three points under the maximum negative velocity, the present LES results are broadly consistent with (4.5).

6.1. Separation bubble

It is of interest to discuss some other features of the separation bubble flows of both Perry & Fairlie (Reference Perry and Fairlie1975) and Patrick (Reference Patrick1987). As discussed previously, the case of Perry and Fairlie has a mild APG combined with a strong FPG, and its $C_{f}$ from the LES shows a strong streamwise asymmetry in figure 8. This behaviour can be observed in the bubble shape, as shown in figure 16(a), where we find the bubble is not perfectly symmetric, being slightly lifted towards the downstream part of the flow domain. Physically, it is reasonable that the bubble is not symmetric owing to the unsymmetrical pressure gradient. Interestingly, this feature is not present in all cases. For the Patrick flow, the $C_{f}(x)$ variation of figure 14, while not perfectly symmetric, is more so than in the case of Perry and Fairlie. The bubble shape is also approximately symmetric. Considering the sharp pressure gradient near separation, a rather large bubble is generated. In figure 16, experiment and LES look reasonably similar near and inside the separation bubble.

Figure 16.

Comparison of separation bubble shape. (a) LES for the case of Perry & Fairlie (Reference Perry and Fairlie1975); (b) experiment by Perry & Fairlie (Reference Perry and Fairlie1975); (c) LES for the case of Patrick (Reference Patrick1987); (d) experiment by Patrick (Reference Patrick1987).

Figure 17.

Effect of Reynolds number: ——, $Re_{{\it\theta}}^{in}=2000$ ; – – – –, $Re_{{\it\theta}}^{in}=8000$ ; — ⋅ —, $Re_{{\it\theta}}^{in}=16\,000$ ; — ⋅ ⋅ —, $Re_{{\it\theta}}^{in}=10^{5}$ ; ▫, from experimental data of Perry & Fairlie (Reference Perry and Fairlie1975). (a)  $C_{f}$ versus $x/{\it\delta}_{0}$ ; (b)  $\overline{h_{0}^{+}}$ versus $x/{\it\delta}_{0}$ ; (c)  ${\it\beta}$ versus $x/{\it\delta}_{0}$ .

Another interesting effect is the change of bubble size due to Reynolds number. The case of Perry and Fairlie was extended to three further LES with $Re_{{\it\theta}}^{in}=8000$ , 16 000 and $10^{5}$ . The same numerical set-up as in case $C1$ was used. In figure 17(a), the $C_{f}(x)$ variation is qualitatively similar for all cases. As $Re_{{\it\theta}}^{in}$ increases, the values of $C_{f}$ before separation and after reattachment reduce substantially at a given $x$ , consistent with the effect of increasing Reynolds number on an attached flow TBL. In contrast, inside the separation bubble, differences in $C_{f}(x)$ are relatively smaller between the various cases, suggesting an interior bubble flow that is only weakly dependent on Reynolds number over a broad range.

To compare the wall-model behaviour in cases with different Reynolds number, the variation of $\overline{h_{0}^{+}}$ versus $x/{\it\delta}_{0}$ is shown in figure 17(b) for the Perry–Fairlie flow. The quantity $h_{0}^{+}(x,y,t)$ is dynamically calculated at every wall point. Here, its mean diagnostic is defined as

The average of ${\it\eta}_{0}$ is implemented in both the spanwise direction and in time. It is interesting that $\overline{h_{0}^{+}}$ does not approach zero even near the separation and reattachment points. This is reasonable as we would expect that ${\it\eta}_{0}(x,y,t)$ , which is proportional the magnitude of the skin-friction vector, would be zero only at isolated critical points. Since we have fixed relation $h^{+}=0.68/0.18h_{0}^{+}$ , the behaviour of $h^{+}$ is not shown. The variation of $\overline{h_{0}^{+}}(x)$ for the Patrick flow is similar to that of figure 17(b).

An issue of some interest is the effect of the boundary condition (2.19) used presently for the wall-normal velocity. For the LES of the Perry–Fairlie flow, we find that at the virtual wall, $\text{Prob}:\{|\tilde{w}|_{h_{0}}/\tilde{q}|_{h_{0}}>0.05\}\approx 0.07$ , $\text{Prob}:\{|\tilde{w}|_{h_{0}}/\tilde{q}|_{h_{0}}\}>0.1\approx 0.038$ and $\text{Prob}:\{|\tilde{w}|_{h_{0}}/\tilde{q}|_{h_{0}}>0.2\}\approx 0.02$ , while for the Patrick flow the corresponding results are $\text{Prob}:\{|\tilde{w}|_{h_{0}}/\tilde{q}|_{h_{0}}>0.05\}\approx 0.097$ , $\text{Prob}:\{|\tilde{w}|_{h_{0}}/\tilde{q}|_{h_{0}}>0.1\}\approx 0.061$ and $\text{Prob}:\{|\tilde{w}|_{h_{0}}/\tilde{q}|_{h_{0}}>0.2\}\approx 0.038$ . These values are small but not negligible and are probably associated with local regions of small $|\boldsymbol{{\it\tau}}_{w}|$ within the large-scale separated flow. When, however, (2.19) is replaced with $|\tilde{w}|_{h_{0}}=0$ , we find LES results for example for the $C_{f}(x)$ distribution (not shown) that are little different to those shown presently.

In passing, we note the large-scale Reynolds numbers $Re_{B}\equiv U_{\infty }\,L_{b}/{\it\nu}$ for the present separation-bubble flows. For the Perry–Fairlie flow with $Re_{{\it\theta}}^{in}=2\times 10^{3},8\times 10^{3}$ , $1.6\times 10^{4}$ and $10^{5}$ , we find $Re_{B}=5.3\times 10^{5},2.1\times 10^{6},4.2\times 10^{6}$ and $4.2\times 10^{7}$ , respectively. For the Patrick case with $R_{{\it\theta}}=1.1\times 10^{4}$ , then $Re_{B}=6\times 10^{5}$ . This shows that $Re_{B}$ for the two experiments are almost the same. This is because for the Patrick case, ${\it\theta}$ is six times larger than for the Perry and Fairlie flow, but the bubble length is five times smaller. These values of $Re_{B}$ serve to indicate that we are performing LES at substantial values of the large-scale Reynolds number.

For quantifying different APG flows over a range of Reynolds number, a commonly used metric is the non-dimensional pressure gradient parameter

which is plotted versus $x/{\it\delta}_{0}$ in figure 17(c). Results show that all three cases plotted collapse into a curve, which is consistent with results by Inoue et al. (Reference Inoue, Pullin, Harun and Marusic2013). When the TBL is lifted by the APG, we observe that ${\it\beta}$ in the lowest Reynolds number case first lifts upwards and then reaches separation. This implies that the separation bubble will shrink with increasing Reynolds number, although this effect is very slow. Lastly, we also observe that the difference between $Re_{{\it\theta}}^{in}=8000$ and $Re_{{\it\theta}}^{in}=16\,000$ is smaller than the difference between $Re_{{\it\theta}}^{in}=2000$ and $Re_{{\it\theta}}^{in}=8000$ . The LES for $Re_{{\it\theta}}^{in}=10^{5}$ shows differences in ${\it\beta}(x)$ compared with $Re_{{\it\theta}}^{in}=16\,000$ of order the plot-line thickness and is therefore not shown.

6.2. $h{-}{\it\lambda}$ relation

A correlation in an $h{-}{\it\Lambda}$ plane has been suggested to demarcate the regions of separation where $h=(H-1)/H$ with $H\equiv {\it\delta}^{\ast }/{\it\theta}$ and ${\it\Lambda}\equiv {\it\delta}^{\ast }/{\it\delta}_{99}$ . Previous studies have identified two important lines on $h{-}{\it\Lambda}$ plots. The line $h=1.5{\it\Lambda}$ was discussed by Simpson (Reference Simpson1989), who stated that the data of Perry & Schofield (Reference Perry and Schofield1973) supported the hypothesis that this characterized a path for detaching flow. Another line is $h=1/(2-{\it\Lambda})$ proposed by Sanborn & Kline (Reference Sanborn and Kline1961). They suggested that at a given ${\it\Lambda}$ , $h$ greater than this value would indicate appreciable intermittent backflow. For a more detailed discussion of the $h{-}{\it\Lambda}$ plane, see Simpson (Reference Simpson1989).

Figure 18.

Comparison of mean velocity profile parameters. In (a) and (b), ——, Perry and Schofield correction; – – – –, Sanborn and Kline relation. In (a) and (c), ▪ with line for LES results of case $F1$ . (a)  $h{-}{\it\Lambda}$ plot for case by Perry & Fairlie (Reference Perry and Fairlie1975). — ⋅ — for $C_{f}$ ; horizontal line for $C_{f}=0$ ; (b)  $h{-}{\it\Lambda}$ plot for case by Patrick (Reference Patrick1987). ▫, experimental data; — ⋅ —, LES result; (c) plot of $h(2-{\it\Lambda})$ versus $\overline{{\it\tau}_{u,x}}$ . Horizontal line for $h(2-{\it\Lambda})=1$ and vertical line for $\overline{{\it\tau}_{w,x}}=0$ .

Figure 18(a) plots the whole path of case $C2$ in the $h{-}{\it\Lambda}$ plane for the Perry & Fairlie (Reference Perry and Fairlie1975) flow. For interpretation purposes, $C_{f}$ is also plotted. The near inflow part, which includes the ZPG portion and subsequent APG part, develops from point A to point B. Here point B is the first point where our LES falls onto the Perry & Schofield (Reference Perry and Schofield1973) line. Point C, corresponding to zero skin friction, which is followed by mean backflow, corresponds closely to the Sanborn–Kline relation, as does the reattachment point D. Between these two points, the LES $h{-}{\it\Lambda}$ plot satisfies the Sanborn–Kline relation criterion. After reattachment, the flow further develops, and the LES $h{-}{\it\Lambda}$ path deviates somewhat from the Perry and Schofield correlation. The recovery of the flow near outflow to the ZPG state can be seen where the LES $h{-}{\it\Lambda}$ path almost closes near point A.

Figure 18(b) compares measurements by Patrick with the present LES. Note that points E and F, which are the separation point and the reattachment points, respectively, agree well with the Sanborn–Kline relation and again the LES $h{-}{\it\Lambda}$ path satisfies the Sanborn–Kline relation in the separated flow region. Although LES cannot exactly reproduce the experimental $h{-}{\it\Lambda}$ path, both are close to the Perry and Schofield correlation. A plot of $h(2-{\it\Lambda})$ versus $\overline{{\it\tau}_{w,x}}$ is provided in figure 18(c) to illustrate the interaction points between LES and Sanborn–Kline relation.