2.1. Overview

Figure 1.

Schematic of filter kernel treatment near walls in WRLES, WMLES and LES based on the volume filtering presented here (a–c). In WRLES, the filter is refined and rescaled to maintain unity with decreasing wall distance. In WMLES, the filter is truncated without changing its size and rescaled with decreasing wall distance. In the volume-filtering LES, the filter kernel is truncated without rescaling with decreasing wall distance. Filtered quantities are also extended a short distance into the solid (red point) so long as the kernel overlaps with the fluid region (shaded red region).

Figure 1. Long description

Panel A: A line graph shows the filter kernel treatment in WRLES. The horizontal axis represents increasing wall distance, and the vertical axis represents the filter kernel. The green lines indicate that the filter is refined and rescaled to maintain unity with decreasing wall distance. Panel B: A line graph illustrates the filter kernel treatment in WMLES. The horizontal axis represents increasing wall distance, and the vertical axis represents the filter kernel. The blue lines show that the filter is truncated without changing its size and rescaled with decreasing wall distance. Panel C: A line graph depicts the filter kernel treatment in volume-filtering LES. The horizontal axis represents increasing wall distance, and the vertical axis represents the filter kernel. The red lines indicate that the filter kernel is truncated without rescaling with decreasing wall distance. Filtered quantities are extended a short distance into the solid (red point) so long as the kernel overlaps with the fluid region (shaded red region).

Before presenting the mathematical details, it is worthwhile starting by emphasising the main differences between different filtering methods and the connection with IB methods.

To help with this discussion, we show in figure 1 the qualitative variation of the filter kernel with wall distance in three methods: WRLES (top panel), WMLES with the treatment of Bose & Moin (Reference Bose and Moin2014) (middle panel) and the volume-filtering LES we present here (bottom panel). First, note that, away from the wall, the filter shape and size, denoted by $\delta _{\kern-1pt f}$ , may be taken as identical in all three methods. These methods differ only in their treatment of the wall region. In WRLES, the filter, be it either explicit or implicit, is refined to a DNS resolution close to walls. That is $\delta _{\kern-1pt f}$ decreases approaching a wall, such that $\delta _{\kern-1pt f}^+\lesssim 1$ within the boundary layer, where the ‘ $+$ ’ notation refers to non-dimensionalisation with inner wall units using combinations of the friction velocity $u_\tau$ and kinematic viscosity $\nu$ . This ensures that near-wall turbulent flow structures are well resolved in WRLES. Shrinking the filter support closer to the wall requires rescaling the filter to ensure that it remains unitary, even though its shape is unchanged. In WMLES, the filter size is typically uniform such that $\delta _{\kern-1pt f}^+=O(10^2)$ everywhere. Because of this, the filter may extend beyond the fluid region at distances less than $\delta _{\kern-1pt f}/2$ from a wall. Bose & Moin (Reference Bose and Moin2014) suggest truncating the portion of the filter that falls inside the solid, thereby altering the filter shape, and rescaling it to ensure that it remains unitary. The consequences of this treatment are discussed in detail in Bose & Moin (Reference Bose and Moin2014).

In the volume-filtering LES we present here, we also maintain a uniform filter size, with a typical size $\delta _{\kern-1pt f}^+=O(10^2)$ , and truncate the part of the kernel that falls within the solid region. However, there are two essential differences from the approach of Bose & Moin (Reference Bose and Moin2014). First, we do not rescale truncated filters. Instead, we introduce a fluid volume fraction $\alpha _{\kern-1pt f}$ to account for the deficit of mass and momentum near a wall. This quantity represents the fraction of the fluid region among the total region (solid and fluid) covered by the filter. For a planar wall, $\alpha _{\kern-1pt f}=1/2$ at the wall since exactly half of the region covered by the filter is fluid. Away from walls, $\alpha _{\kern-1pt f}=1$ . Second, we extend our definitions of filtered quantities inside the solid region. The red point in the bottom panel of figure 1 illustrates this aspect. The filtered velocity at this point represents the average velocity within the portion of the filter that falls in the fluid region (portion shaded in red). In general, filtered fluid quantities may be non-zero at points inside the solid but within $\delta _{\kern-1pt f}/2$ from a wall.

Compared with other WMLES methods, volume-filtering LES has two significant advantages: (i) it eliminates commutation errors and (ii) turns boundary conditions, such as the no-slip condition, into volumetric forcing, effectively leading to an IB method. In contrast, the WRLES and WMLES discussed above must contend with commutation errors. Since the filter size and/or shape vary with wall distance, filtering and derivative operators are no longer commutative (Germano Reference Germano, Moreau and Rodi2002; Gullbrand Reference Gullbrand2003; Sagaut Reference Sagaut2006; Klein & Germano Reference Klein and Germano2020). This leads to additional terms in the filtered transport equations, which can be of the order of the subgrid terms (Ghosal & Moin Reference Ghosal and Moin1995; Jordan Reference Jordan1999; Gullbrand Reference Gullbrand2003; Vanella et al. Reference Vanella, Piomelli and Balaras2008; Klein & Germano Reference Klein and Germano2020). These commutation terms are ignored in most WRLES and WMLES methods. Volume-filtering and differentiation operations are not commutative either. The terms arising from the non-commutative operations are part of what forms the IB bodyforce.

2.2. Derivation

We now present the mathematical details underpinning the volume-filtering framework and discuss the commutation terms, their connection with the IB bodyforce and the different closure problems. This treatment follows our earlier work in Dave et al. (Reference Dave, Herrmann and Kasbaoui2023) and Kasbaoui & Herrmann (Reference Kasbaoui and Herrmann2025), where we first introduced the volume-filtering IB method. Unlike in Dave et al. (Reference Dave, Herrmann and Kasbaoui2023), we make no assumptions on the relative size of the filter kernel $\delta _{\kern-1pt f}$ with respect to the flow structures.

In the following, we assume that the fluid is Newtonian, has constant density $\rho _{\kern-1pt f}$ and constant viscosity $\mu _{\kern-1pt f}$ . Thus, the equations of motion for this fluid are dictated by the incompressible Navier–Stokes equations

(2.1) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}_{\kern-1pt f}=0, \\[-28pt] \nonumber \end{align}

(2.2) \begin{align} \rho _{\kern-1pt f}\left (\frac {\partial \boldsymbol{u}_{\kern-1pt f}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\boldsymbol{u}_{\kern-1pt f} \boldsymbol{u}_{\kern-1pt f}\right )\right )=\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f}, \\[0pt] \nonumber \end{align}

where $\boldsymbol{u}_{\kern-1pt f}$ is the unfiltered fluid velocity, $\boldsymbol{\mathsf{\tau }}_{\kern-2pt f}=-p \unicode{x1D644} + \mu _{\kern-1pt f} (\boldsymbol{\nabla }\boldsymbol{u}_{\kern-1pt f} + \boldsymbol{\nabla }\boldsymbol{u}_{\kern-1pt f}^{\text{T}} )$ is the total stress tensor ( $\unicode{x1D644}$ and $(.)^{T}$ denote the identity matrix and transpose, respectively) and $p$ is the fluid pressure.

To derive volume-filtered equations, we consider a filter kernel $\mathcal{G}(\boldsymbol{y})$ for $\boldsymbol{y}\in \textit{Re} ^3$ with size $\delta _{\kern-1pt f}$ . The filter kernel must satisfy two fundamental conditions

(2.3) \begin{align} \iiint _{\boldsymbol{y}\in \textit{Re} ^3}\mathcal{G}(\boldsymbol{y})\text{d}V=1, \\[-28pt] \nonumber \end{align}

(2.4) \begin{align} \mathcal{G}(-\boldsymbol{y})=\mathcal{G}(\boldsymbol{y}). \\[0pt] \nonumber \end{align}

The first dictates that the filter is unitary while the second requires it to be symmetric. Although not strictly required for the derivation, it is also convenient to take a compact filter kernel, i.e. one that verifies the condition

(2.5) \begin{align} \mathcal{G}(\boldsymbol{y})=0\quad \text{if}\:||\boldsymbol{y}||\geqslant \delta _{\kern-1pt f}/2. \end{align}

We define the fluid volume fraction $\alpha _{\kern-1pt f}$ and filtered fluid velocity $\overline {\boldsymbol{u}}_{\kern-1pt f}$ as follows:

(2.6) \begin{align} \alpha _{\kern-1pt f}(\boldsymbol{x},t) = \iiint _{\boldsymbol{y}\in \textit{Re} ^3}\mathbb{I}_f(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}V, \\[-28pt] \nonumber \end{align}

(2.7) \begin{align} \alpha _{\kern-1pt f}(\boldsymbol{x},t)\overline {\boldsymbol{u}}_{\kern-1pt f}(\boldsymbol{x},t)=\iiint _{\boldsymbol{y}\in \textit{Re} ^3}\mathbb{I}_f(\boldsymbol{y},t)\boldsymbol{u}_{\kern-1pt f}(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}V. \\[0pt] \nonumber \end{align}

Here, $\mathbb{I}_f(\boldsymbol{y},t)$ is the fluid-phase-indicator function whose value is 1 if $\boldsymbol{y}$ is in the fluid at time $t$ and 0 otherwise. The function $\mathbb{I}_f$ truncates the filtering region, as schematised in the bottom panel of figure 1. From (2.6), the volume fraction may be interpreted as the filtered indicator function. That is, $\mathbb{I}_f$ is a sharp and discontinuous field, whereas $\alpha _{\kern-1pt f}$ is a smooth field that diffuses the solid–fluid interface over a band of width $\delta _{\kern-1pt f}$ . Other fluid quantities may be filtered in the same manner as the fluid velocity in (2.7).

As previously discussed, filtering and differentiation operations are not commutative. This results from the truncation of the filtering region by the phase-indicator function $\mathbb{I}_f$ . Still, we can show that the following identities hold exactly (Dave et al. Reference Dave, Herrmann and Kasbaoui2023):

(2.8) \begin{align} \alpha _{\kern-1pt f}(\boldsymbol{x},t)\overline {\boldsymbol{\nabla }\boldsymbol{\phi }}(\boldsymbol{x},t)=\boldsymbol{\nabla }\left (\alpha _{\kern-1pt f}\overline {\boldsymbol{\phi }}\right )-\iint _{\boldsymbol{y}\in S_w}\boldsymbol{n}\boldsymbol{\phi }(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S, \\[-28pt] \nonumber \end{align}

(2.9) \begin{align} \alpha _{\kern-1pt f}(\boldsymbol{x},t)\overline {\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\phi }}(\boldsymbol{x},t)=\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\alpha _{\kern-1pt f}\overline {\boldsymbol{\phi }}\right )-\iint _{\boldsymbol{y}\in S_w}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\phi }(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S, \\[-28pt] \nonumber \end{align}

(2.10) \begin{align} \alpha _{\kern-1pt f}(\boldsymbol{x},t)\overline {\frac {\partial \boldsymbol{\phi }}{\partial t}}(\boldsymbol{x},t)=\frac {\partial }{\partial t}\left (\alpha _{\kern-1pt f}\overline {\boldsymbol{\phi }}\right )+\iint _{\boldsymbol{y}\in S_w}(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{u}_w)\boldsymbol{\phi }(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S, \\[0pt] \nonumber \end{align}

where $\boldsymbol{\phi }(\boldsymbol{x})$ is a scalar or vector property of the fluid, $\boldsymbol{n}$ is a unitary vector normal to the wall and pointing from the solid to the fluid, $\boldsymbol{u}_w$ is the local wall velocity and $S_w$ denotes all walls. It is also possible to rearrange the equations above to get exact expressions of the commutation terms

(2.11) \begin{align} \overline {\boldsymbol{\nabla }\boldsymbol{\phi }} - \boldsymbol{\nabla }\overline {\boldsymbol{\phi }} = \frac {1}{\alpha _{\kern-1pt f}}\left ((\boldsymbol{\nabla }\alpha _{\kern-1pt f}) \overline {\boldsymbol{\phi }} -\iint _{\boldsymbol{y}\in S_w}\boldsymbol{n}\boldsymbol{\phi }(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S\right )\!, \\[-28pt] \nonumber \end{align}

(2.12) \begin{align} \overline {\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\phi }}-\boldsymbol{\nabla }\boldsymbol{\cdot }\overline {\boldsymbol{\phi }} = \frac {1}{\alpha _{\kern-1pt f}}\left ((\boldsymbol{\nabla }\alpha _{\kern-1pt f})\boldsymbol{\cdot }\overline {\boldsymbol{\phi }}-\iint _{\boldsymbol{y}\in S_w}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\phi }(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S\right )\!, \\[-28pt] \nonumber \end{align}

(2.13) \begin{align} \overline {\frac {\partial \boldsymbol{\phi }}{\partial t}}-\frac {\partial \overline {\boldsymbol{\phi }}}{\partial t} = \frac {1}{\alpha _{\kern-1pt f}}\left ( \left (\frac {\partial \alpha _{\kern-1pt f}}{\partial t}\right ) \overline {\boldsymbol{\phi }} +\iint _{\boldsymbol{y}\in S_w}(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{u}_w)\boldsymbol{\phi }(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S\right )\!. \\[0pt] \nonumber \end{align}

To obtain the volume-filtered transport equations, we apply the filtering operation to the transport equations (2.1) and (2.2). With help of identities (2.8)–(2.10), we get

(2.14) \begin{align} \frac {\partial \alpha _{\kern-1pt f}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\alpha _{\kern-1pt f}\overline {\boldsymbol{u}}_{\kern-1pt f})=0, \\[-28pt] \nonumber \end{align}

(2.15) \begin{align} \rho _{\kern-1pt f}\left (\frac {\partial }{\partial t}\left (\alpha _{\kern-1pt f}\overline {\boldsymbol{u}}_{\kern-1pt f}\right )+\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\alpha _{\kern-1pt f}\overline {\boldsymbol{u}_{\kern-1pt f}\boldsymbol{u}_{\kern-1pt f}}\right )\right )&=\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\alpha _{\kern-1pt f}\overline {\boldsymbol{\mathsf{\tau }}}_{\kern-1pt f}\right ) \nonumber \\& \quad -\iint _{\boldsymbol{y}\in S_w}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f}(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S. \\[0pt] \nonumber \end{align}

The filtered stress tensor $\overline {\boldsymbol{\mathsf{\tau }}}_{\kern-1pt f}$ in the filtered momentum (2.15) requires special care. We decompose this tensor into two contributions, $\overline {\boldsymbol{\mathsf{\tau }}}_{\kern-1pt f}=\overline {\boldsymbol{\mathsf{\tau }}}_{\kern-1pt f}^R + \unicode{x1D64D}_\mu$ , where each of these reads

(2.16) \begin{eqnarray} \overline {\boldsymbol{\mathsf{\tau }}}_{\kern-1pt f}^R&=&-\overline {p} \unicode{x1D644} + \mu _{\kern-1pt f}\left (\boldsymbol{\nabla }\overline {\boldsymbol{u}}_{\kern-1pt f} + \boldsymbol{\nabla }\overline {\boldsymbol{u}}_{\kern-1pt f}^T-\frac {2}{3}\left (\boldsymbol{\nabla }\boldsymbol{\cdot }\overline {\boldsymbol{u}}_{\kern-1pt f}\right ) \unicode{x1D644}\right )\!, \end{eqnarray}

(2.17) \begin{eqnarray} \unicode{x1D64D}_\mu &=&\mu _{\kern-1pt f} \left (\overline {\boldsymbol{\nabla }\boldsymbol{u}_{\kern-1pt f}+\boldsymbol{\nabla }\boldsymbol{u}_{\kern-1pt f}^T}\right ) - \mu _{\kern-1pt f}\left (\boldsymbol{\nabla }\overline {\boldsymbol{u}}_{\kern-1pt f}+\boldsymbol{\nabla }\overline {\boldsymbol{u}}_{\kern-1pt f}^T-(2/3)(\boldsymbol{\nabla }\boldsymbol{\cdot }\overline {\boldsymbol{u}}_{\kern-1pt f}) \unicode{x1D644}\right )\!. \end{eqnarray}

The tensors $\overline {\boldsymbol{\mathsf{\tau }}}_{\kern-1pt f}^R$ and $ \unicode{x1D64D}_{\mu }$ as the so-called resolved fluid stress tensor and residual viscous stress tensor, respectively. While both tensors are closed, $ \unicode{x1D64D}_{\mu }$ has a non-trivial form and is poorly understood. Using (2.11), (2.12) and the no-slip condition $\boldsymbol{u}_{\kern-1pt f}|_w=\boldsymbol{u}_w$ , we get an alternative expression for $ \unicode{x1D64D}_{\mu }$

(2.18) \begin{eqnarray} \unicode{x1D64D}_\mu &=&\frac {\mu _{\kern-1pt f}}{\alpha _{\kern-1pt f}}\Biggl ( (\boldsymbol{\nabla }\alpha _{\kern-1pt f}) \overline {\boldsymbol{u}}_{\kern-1pt f} + \overline {\boldsymbol{u}}_{\kern-1pt f} (\boldsymbol{\nabla }\alpha _{\kern-1pt f}) - (2/3)(\boldsymbol{\nabla }\alpha \boldsymbol{\cdot }\overline {\boldsymbol{u}}_{\kern-1pt f}) \unicode{x1D644}\nonumber \\&& -\iint _{\boldsymbol{y}\in S_w} \big (\boldsymbol{n}\boldsymbol{u}_w+\boldsymbol{u}_w\boldsymbol{n}-(2/3)(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{u}_w) \unicode{x1D644} \kern1pt\big )\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S \Biggr ). \end{eqnarray}

This symmetric tensor vanishes quickly away from the fluid–solid interface. Close to walls, $ \unicode{x1D64D}_{\mu }$ scales with the difference between the filtered fluid velocity at the wall $\overline {\boldsymbol{u}}_{\kern-1pt f}|_w$ and the wall velocity itself $\boldsymbol{u}_w$ . We refer to this as the wall-slip velocity

(2.19) \begin{align} \boldsymbol{u}_{\textit{slip}}=\overline {\boldsymbol{u}}_{\kern-1pt f}|_w- \boldsymbol{u}_w. \end{align}

As with other LES methods, the convective term $\overline {\boldsymbol{u}_{\kern-1pt f}\boldsymbol{u}_{\kern-1pt f}}$ in (2.15) presents a closure problem. Following standard LES strategies, we define the SFS stress tensor

(2.20) \begin{align} \boldsymbol{\mathsf{\tau }}_{\kern-1pt \textit{sfs}}=\overline {\boldsymbol{u}_{\kern-1pt f}\boldsymbol{u}_{\kern-1pt f}}-\overline {\boldsymbol{u}}_{\kern-1pt f}\overline {\boldsymbol{u}}_{\kern-1pt f}. \end{align}

Introducing this tensor in (2.15) and using the product rule on the first term on the right-hand side, we obtain the following form of the volume-filtered conservation equations:

(2.21) \begin{align} \frac {\partial \alpha _{\kern-1pt f}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\alpha _{\kern-1pt f}\overline {\boldsymbol{u}}_{\kern-1pt f})=0, \\[-28pt] \nonumber \end{align}

(2.22) \begin{align} \rho _{\kern-1pt f}\left (\frac {\partial }{\partial t}\left (\alpha _{\kern-1pt f}\overline {\boldsymbol{u}}_{\kern-1pt f}\right )+\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\alpha _{\kern-1pt f}\overline {\boldsymbol{u}}_{\kern-1pt f}\overline {\boldsymbol{u}}_{\kern-1pt f}\right )\right ) &=& \alpha _{\kern-1pt f}\boldsymbol{\nabla }\boldsymbol{\cdot }\big (\overline {\boldsymbol{\mathsf{\tau }}}^R_{\kern-2pt f}+ \unicode{x1D64D}_{\mu }-\rho _{\kern-1pt f} \boldsymbol{\mathsf{\tau }}_{\kern-1pt \textit{sfs}}\big ) +\boldsymbol{F}_{\kern-1pt \textit{IB}}, \\[0pt] \nonumber \end{align}

where $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ is a bodyforce that reads

(2.23) \begin{align} \boldsymbol{F}_{\kern-1pt \textit{IB}}(\boldsymbol{x},t) = -\iint _{\boldsymbol{y}\in S_w}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f}(\boldsymbol{y},t)\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S + (\boldsymbol{\nabla }\alpha _{\kern-1pt f})\boldsymbol{\cdot }\big (\overline {\boldsymbol{\mathsf{\tau }}}^R_{\kern-2pt f}+ \unicode{x1D64D}_{\mu }-\rho _{\kern-1pt f} \boldsymbol{\mathsf{\tau }}_{\kern-1pt \textit{sfs}}\big ) . \end{align}

Leveraging the identity in (2.8) with $\boldsymbol{\phi }=\boldsymbol{I}$ , it is possible to approximate $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ in the following manner:

(2.24) \begin{align} \boldsymbol{F}_{\kern-1pt \textit{IB}}(\boldsymbol{x},t)=-\iint _{\boldsymbol{y}\in S_w}\boldsymbol{n}\boldsymbol{\cdot }\big (\boldsymbol{\mathsf{\tau }}_{\kern-2pt f}-\overline {\boldsymbol{\mathsf{\tau }}}^R_{\kern-2pt f}- \unicode{x1D64D}_{\mu }+\rho _{\kern-1pt f} \boldsymbol{\mathsf{\tau }}_{\kern-1pt \textit{sfs}}\big )\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S + O\left (\delta _{\kern-1pt f}^2\right )\!. \end{align}

This term represents a force density exerted by the immersed solid on the fluid. It is non-zero only in close vicinity to the fluid–solid interface. With a compact filter, this is exactly a band of size $\delta _{\kern-1pt f}$ centred on the interface. The bodyforce $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ arises from stresses on the fluid–solid interface in this filtered framework.

From a computational perspective, the term $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ may be thought of as the IB forcing since it serves to impose boundary conditions on the fluid–solid interface. As we have shown in Dave et al. (Reference Dave, Herrmann and Kasbaoui2023), $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ imposes the no-slip boundary condition in the DNS limit, i.e. when $\delta _{\kern-1pt f}\rightarrow 0$ .

As anticipated, the filtered momentum (2.22) requires closure. The unclosed terms here are (i) the SFS tensor $\boldsymbol{\mathsf{\tau }}_{\kern-1pt \textit{sfs}}$ and (ii) the IB bodyforce $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ . The residual viscous stress tensor $ \unicode{x1D64D}_\mu$ is closed, albeit that it has a non-trivial dependence on the filtered quantities that ought to be clarified. Since there have been numerous studies devoted to closing $\boldsymbol{\mathsf{\tau }}_{\kern-1pt \textit{sfs}}$ , the rest of this work focuses on $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ and $ \unicode{x1D64D}_\mu$ , which represent terms unique to this method.

2.3. Modelling the immersed boundary bodyforce

Closing the IB bodyforce $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ amounts to formulating a wall model since the wall stress $(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f})|_w$ is the only unclosed term in expressions (2.23) and (2.24).

In principle, traditional wall models, such as algebraic wall models, approximate boundary conditions (Schumann Reference Schumann1975; Piomelli et al. Reference Piomelli, Ferziger, Moin and Kim1989), and thin boundary-layer equations (Bose & Park Reference Bose and Park2018), could be adapted to this framework. These methods are typically formulated to predict the wall stress $(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f})|_w$ from the filtered velocity $\overline {\boldsymbol{u}}_{\kern-1pt f}$ and its gradients close to walls. To use these models in this framework, one would insert the wall stress $(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f})|_w$ determined by these models in $\boldsymbol{F}_{\kern-1pt \textit{IB}}$ via expression (2.23) or (2.24).

In this study, we propose an alternative method to wall modelling that is based on modelling the wall-slip velocity $\boldsymbol{u}_{\textit{slip}}=\overline {\boldsymbol{u}}_{\kern-1pt f}|_w-\boldsymbol{u}_w$ , i.e. the difference between the filtered velocity at the solid–fluid interface and the actual wall velocity, rather than wall stress $(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f})|_w$ . Following the treatment in Dave et al. (Reference Dave, Herrmann and Kasbaoui2023), we can show that the wall slip $\boldsymbol{u}_{\textit{slip}}$ and wall stress $(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f})|_w$ are related to one another via

(2.25) \begin{align} \left .(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\mathsf{\tau }}_{\kern-2pt f})\right |_w &= \left .\boldsymbol{n}\boldsymbol{\cdot }\big (\overline {\boldsymbol{\mathsf{\tau }}}^R_{\kern-2pt f}+ \unicode{x1D64D}_{\mu }-\rho _{\kern-1pt f} \boldsymbol{\mathsf{\tau }}_{\kern-1pt \textit{sfs}}\big )\right |_w \nonumber \\& \quad + \rho _{\kern-1pt f}\frac {\alpha _{f,w}}{\varSigma _w}\left [\frac{\text{d}}{\text{d}t}\left (\boldsymbol{u}_w +\boldsymbol{u}_{\textit{slip}}\right )-\left .\boldsymbol{\nabla }\boldsymbol{\cdot }\big (\overline {\boldsymbol{\mathsf{\tau }}}^R_{\kern-2pt f}+ \unicode{x1D64D}_{\mu }-\boldsymbol{\mathsf{\tau }}_{\kern-1pt \textit{sfs}}\big )\right |_w\right ] +O \big(\delta _{\kern-1pt f}^2 \big), \end{align}

where $\alpha _{f,w}$ and $\varSigma _{w}$ are the fluid volume fraction and surface density, respectively, at the wall. Here, the surface density function is

(2.26) \begin{eqnarray} \varSigma (\boldsymbol{x},t) = \iint _{\boldsymbol{y}\in S_w}\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S. \end{eqnarray}

Integrating $\varSigma$ over the entire domain gives the total wall area. For planar walls, $\alpha _{f,w}=1/2$ and $\varSigma _w=O(1/\delta _{\kern-1pt f})$ , the precise value depending on the kernel $\mathcal{G}$ .

In (2.25), the wall-slip velocity $\boldsymbol{u}_{\textit{slip}}$ concentrates the unknown, as it is the only unclosed term. Thus, formulating a model for $\boldsymbol{u}_{\textit{slip}}$ is equivalent to specifying a wall model. This approach is better suited in IB methods, since most of these methods are formulated to impose a Dirichlet condition on the velocity via a volumetric forcing term.

It is important to realise that the no-slip boundary condition, i.e. $\boldsymbol{u}_{\textit{slip}}=0$ , applies only when the filter size is very small. For wall-bounded turbulence, this requires $\delta _{\kern-1pt f}^+\lt 1$ , which we refer to as the DNS limit (Dave et al. Reference Dave, Herrmann and Kasbaoui2023; Kasbaoui & Herrmann Reference Kasbaoui and Herrmann2025).

Figure 2.

The filtered fluid velocity at the wall represents an average velocity of the flow in the region under the filter kernel centred at the wall (marked in grey). This leads to significant slip velocity at filter sizes $\delta _{\kern-1pt f}^+=O(100)$ .

For larger filter sizes, the no-slip condition ( $\boldsymbol{u}_{\textit{slip}}=0$ ) is no longer valid. That is because the filtered velocity at the wall $\overline {\boldsymbol{u}}_{\kern-1pt f}|_w$ represents the average fluid velocity in a volume of size $\delta _{\kern-1pt f}$ centred at the wall. Figure 2 shows a schematic of the averaging volume and the flow structures that may be part of this region. For vanishingly small filter sizes ( $\delta _{\kern-1pt f}\rightarrow 0$ ), the averaging volume collapses onto a point, which leads to $\overline {\boldsymbol{u}}_{\kern-1pt f}|_w\rightarrow \boldsymbol{u}_w$ . Conversely, increasing the filter size increases the slip as the averaging volume used to compute $\overline {\boldsymbol{u}}_{\kern-1pt f}|_w$ covers increasingly more of the viscous, buffer and logarithmic sublayers.

The residual viscous stress tensor and the slip velocity are also closely connected. Considering expression (2.18) at the wall, one can show

(2.27) \begin{eqnarray} \left .{ \unicode{x1D64D}_\mu }\right |_w &=&\frac {\mu _{\kern-1pt f}}{\alpha _{f,w}} \iint _{\boldsymbol{y}\in S_w} \big (\boldsymbol{n}\boldsymbol{u}_{\textit{slip}}+\boldsymbol{u}_{\textit{slip}}\boldsymbol{n}-(2/3)(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{u}_{\textit{slip}})\boldsymbol{I}\big )\mathcal{G}(\boldsymbol{x}-\boldsymbol{y})\text{d}S. \end{eqnarray}

Thus, $ \unicode{x1D64D}_\mu$ scales with the slip velocity. Since the latter is significant for filter sizes $\delta _{\kern-1pt f}^+=O(100)$ , $ \unicode{x1D64D}_\mu$ is also significant at those filter sizes and must be accounted for correctly.

3.1. Overview of the filtering procedure

To filter the DNS data, we consider a kernel that is expressed as the product of three one-dimensional kernels

(3.1) \begin{align} \mathcal{G}(x,y,z)=\mathcal{G}_1(x)\mathcal{G}_1(y)\mathcal{G}_1(z). \end{align}

where $x$ , $y$ and $z$ are the streamwise, wall-normal and spanwise coordinates, respectively. Since the three-dimensional kernel $\mathcal{G}$ must be symmetric and unitary, the one-dimensional kernels $\mathcal{G}_1$ in (3.1) must also satisfy the same conditions.

The data we present throughout this section result from filtering with the cosine kernel

(3.2) \begin{align} \mathcal{G}_1(r)=\begin{cases} \frac {\pi }{2\delta _{\kern-1pt f}}\cos {\left (\frac {\pi r}{\delta _{\kern-1pt f}}\right )} & |r|\lt \delta _{\kern-1pt f}/2 ,\\[4pt] 0 & |r|\geqslant \delta _{\kern-1pt f}/2 .\end{cases} \end{align}

We also consider other kernels in § 5 and show that there are no qualitative differences.

We filter the DNS data using filter sizes from $\delta _{\kern-1pt f}^+=50$ up to $300$ at regular intervals of $50$ . To this end, we compute the volume-filtered quantities by direct application of the convolution (see (2.6) and (2.7)) on a uniform grid such that $\Delta x=\Delta y=\Delta z=\delta _{\kern-1pt f}/16$ . Note that ( $\Delta x$ , $\Delta y$ , $\Delta z$ ) are the grid spacings in our post-processing, not to be confused with the DNS grid spacings ( $\Delta x_{\textit{DNS}}$ , $\Delta y_{\textit{DNS}}$ , $\Delta z_{\textit{DNS}}$ ). We chose our post-processing resolution based on prior work in Dave et al. (Reference Dave, Herrmann, Brady and Kasbaoui2025), where we showed that numerical filtering errors become negligible with $\Delta x \leqslant \delta _{\kern-1pt f}/16$ . For $\delta _{\kern-1pt f}^+\geqslant 200$ , the post-processing grid spacing is coarser than the DNS grid spacings since $\Delta x^+=\delta _{\kern-1pt f}^+/16=12.5$ , whereas $\Delta x^+_{\textit{DNS}}\sim 12$ , $\Delta z^+_{\textit{DNS}}\simeq 6$ and $\Delta y^+_{\textit{DNS}}$ varies from $\sim 0.5$ to $\sim 10$ . For $\delta _{\kern-1pt f}^+=50$ and $\delta _{\kern-1pt f}^+=100$ , the post-processing grid spacing falls below the DNS grid spacings, since $\Delta x^+=3.125$ and $\Delta x^+=6.25$ at $\delta _{\kern-1pt f}^+=50$ and 100, respectively. In these cases, we use linear interpolations to evaluate the DNS data on the post-processing grid. However, to avoid contamination from interpolation errors we do not consider filter sizes smaller than $\delta _{\kern-1pt f}^+=50$ .

Further, we extend the post-processing domain boundaries by a distance $\delta _{\kern-1pt f}/2$ away from the top and bottom channel extents, and we place immersed solids in this extended region to represent the channel walls. The fluid-facing boundaries of these immersed solids are what we refer to as the ‘wall’ or ‘immersed boundary’ for the remainder of this manuscript. It is also important to note that, in this a priori analysis, the filtered quantities are computed at collocated points at cell centres as the DNS data available to us are also collocated at cell corners. In an a posteriori analysis of the IBMLES framework, filtered quantities would be computed on a staggered grid.

We start by showing qualitatively the effect of volume filtering using a coarse filter at $\delta _{\kern-1pt f}^+=300$ . Figure 3 shows instantaneous snapshots of the unfiltered streamwise velocity $u^+_{\kern-1pt f}=u_{\kern-1pt f}/u_\tau$ , the corresponding superficial and filtered streamwise velocities, $\alpha _{\kern-1pt f}\overline {u}^+_{\kern-2pt f}$ and $\overline {u}^+_{\kern-2pt f}$ respectively, and volume fraction $\alpha _{\kern-1pt f}$ in a wall-normal streamwise plane at an arbitrary time. The fluid volume fraction $\alpha _{\kern-1pt f}$ clearly demarcates the fluid and solid regions where it is equal to 1 and 0, respectively. Close to the wall, there is a transition region of thickness $\delta _{\kern-1pt f}^+=300$ resulting from volume filtering the phase-indicator function. At this coarse filter size, the superficial velocity $\alpha _{\kern-1pt f}\overline {u}^+_{\kern-2pt f}$ and filtered velocity $\overline {u}^+_{\kern-2pt f}$ are devoid of finer scales present in the unfiltered field $u^+_{\kern-1pt f}$ . Yet, they retain many of the large-scale structures seen in $u^+_{\kern-1pt f}$ . Close to the wall, $\alpha _{\kern-1pt f}\overline {u}^+_{\kern-2pt f}$ vanishes quickly, in part due to the decay of $\alpha _{\kern-1pt f}$ premultiplying this quantity. Without $\alpha _{\kern-1pt f}$ premultiplying $\overline {u}^+_{\kern-2pt f}$ , the value is significant at the wall ( $y^+=0$ ), then decreases sharply to zero at position $y^+=-\delta _{\kern-1pt f}^+/2$ into the solid. This illustrates the point that volume-filtered quantities may be significant up to a distance $\delta _{\kern-1pt f}/2$ into the immersed solid.

Figure 3.

Instantaneous snapshots of the streamwise unfiltered, velocity $u^+_{\kern-1pt f}(\boldsymbol{x})$ , superficial fluid velocity $\alpha _{\kern-1pt f}\overline {u}^+_{\kern-2pt f}(\boldsymbol{x})$ , filtered velocity $\overline {u}^+_{\kern-2pt f}(\boldsymbol{x})$ and volume fraction $\alpha _{\kern-1pt f}(\boldsymbol{x})$ for $\delta ^+_f=300$ (a–d). Large-scale structures remain in the filtered fields while fine-scale structures are filtered out. The $\overline {u}^+_{\kern-2pt f}(\boldsymbol{x})$ field shows clearly that filtered quantities can be non-zero a distance of $\delta _{\kern-1pt f}/2$ into the wall and the $\alpha _{\kern-1pt f}(\boldsymbol{x})$ field indicates fluid and solid regions with a transition region near the wall.

Figure 4.

Fluid volume fraction profiles for (a) channel half-height and (b) near-wall region at filter sizes $\delta _{\kern-1pt f}^+=50$ ( ), $\delta _{\kern-1pt f}^+=100$ ( ), $\delta _{\kern-1pt f}^+=150$ ( ), $\delta _{\kern-1pt f}^+=200$ ( ), $\delta _{\kern-1pt f}^+=250$ ( ) and $\delta _{\kern-1pt f}^+=300$ ( ). The volume fraction smoothly transitions from 0 to 1 over a distance of $\delta _{\kern-1pt f}$ centred at the wall.

The filter size controls the degree to which the wall region is diffused. Figure 4 shows the effect of varying filter size on vertical profiles of fluid volume fraction $\alpha _{\kern-1pt f}$ in the channel’s lower half. Here, $\alpha _{\kern-1pt f}=\alpha _{\kern-1pt f}(y)$ varies with wall-normal distance $y$ only, since the channel walls are flat and immobile. The transition from $\alpha _{\kern-1pt f}=0$ to $\alpha _{\kern-1pt f}=1$ occurs over a distance exactly equal to $\delta _{\kern-1pt f}$ since the kernel has compact support. Since this flow is highly turbulent with $ \textit{Re}_\tau =5200$ , the transition region remains narrow compared with the channel half-height, even at the largest filter size $\delta _{\kern-1pt f}^+=300$ . At the walls ( $y=0$ and $y=2h$ ), $\alpha _{f,w}=1/2$ in all cases, as half of the volume under the filter kernel is occupied by fluid regardless of the filter size. At positions $\delta _{\kern-1pt f}/2\leqslant y\leqslant 2h-\delta _{\kern-1pt f}/2$ , $\alpha _{\kern-1pt f}=1$ identically. Conversely, $\alpha _{\kern-1pt f}=0$ at positions $y\leqslant -\delta _{\kern-1pt f}/2$ and $y\geqslant 2h+\delta _{\kern-1pt f}/2$ .

3.2. Volume-filtered velocity statistics

Figure 5.

Profiles of normalised streamwise velocity in (a) linear and (b) semi-logarithmic scales at filter sizes $\delta _{\kern-1pt f}^+=50$ ( ), $\delta _{\kern-1pt f}^+=100$ ( ), $\delta _{\kern-1pt f}^+=150$ ( ), $\delta _{\kern-1pt f}^+=200$ ( ), $\delta _{\kern-1pt f}^+=250$ ( ), $\delta _{\kern-1pt f}^+=300$ ( ) and unfiltered DNS data ( ). The filtered velocity profiles converge to DNS profile away from the wall while significant slip is seen at the wall. All filtered profiles decrease to 0 at a distance $\delta _{\kern-1pt f}/2$ into the wall.

Figure 5. Long description

Panel A: A line graph shows normalized streamwise velocity profiles in linear scale. The x-axis represents the normalized wall distance (y/h), and the y-axis represents the normalized streamwise velocity (u_f/u_tau). Multiple lines represent different filter sizes, and an unfiltered DNS data line is also present. The filtered velocity profiles converge to the DNS profile away from the wall but show significant slip at the wall. All filtered profiles decrease to 0 at a certain distance into the wall. Panel B: A line graph shows normalized streamwise velocity profiles in semi-logarithmic scale. The x-axis represents the normalized wall distance (y+), and the y-axis represents the normalized streamwise velocity (u_f/u_tau). Similar to Panel A, multiple lines represent different filter sizes, and an unfiltered DNS data line is also present. The trends and convergence patterns are consistent with those observed in Panel A.

Next, we consider the statistics of the volume-filtered fluid velocity. To this end, we average all available flow snapshots from Lee & Moser (Reference Lee and Moser2015) over the streamwise and spanwise directions, and between the channel’s top and bottom halves. We use the notation $\langle \boldsymbol{\cdot }\rangle$ to denote quantities averaged in this manner.

Figures 5(a) and 5(b) show $\langle \overline {u}_{\kern-1pt f}^+\rangle$ profiles in linear and semi-logarithmic scales at filter sizes $\delta _{\kern-1pt f}^+=50$ to 300, alongside the $\langle u_{\kern-1pt f}^+\rangle$ profile from DNS. From $y\sim \delta _{\kern-1pt f}$ and further away from the wall, the mean volume-filtered velocity profiles collapse onto the DNS profile with all filters considered. The $\langle \overline {u}_{\kern-1pt f}^+\rangle$ profiles also show a logarithmic region at distances $y^+\gt \delta _{\kern-1pt f}^+$ .

In contrast, we observe large differences close to the wall. As with the volume fraction field $\alpha _{\kern-1pt f}$ , $\langle \overline {u}_{\kern-1pt f}^+\rangle$ extends beyond the wall and does not vanish until $y=-\delta _{\kern-1pt f}/2$ into the solid. At the wall $y=0$ , the mean slip velocity $\langle u_{\textit{slip}}\rangle =\langle \overline {u}_f\rangle |_{y=0} -\langle u_{\kern-1pt f}\rangle |_{y=0}$ is significant, since $\langle u_{\textit{slip}}\rangle /u_{\tau }=O(10)$ , and increases with $\delta _{\kern-1pt f}$ . From figure 5(b), $\langle \overline {u}_f\rangle$ profiles no longer show the viscous sublayer and buffer regions at these filter sizes. Capturing these regions requires filter sizes of the order of the DNS resolution, i.e. $\delta _{\kern-1pt f}^+\sim 1$ .

Figure 6.

Evolution of (a) the mean streamwise slip and (b) streamwise, (c) wall-normal and (d) spanwise r.m.s. slip fluctuations with filter size. Streamwise slip velocity is significant and shows logarithmic scaling at large $\delta _{\kern-1pt f}^+$ . Streamwise slip velocity fluctuations are approximately $10\,\%$ or less of mean slip values while spanwise and wall-normal fluctuations are negligible.

With filter sizes $\delta _{\kern-1pt f}^+\geqslant 50$ , it is clear that the no-slip condition is no longer valid for the volume-filtered velocity. To further clarify this point, we report the effect of varying filter size on the mean slip and root-mean-square (r.m.s.) slip fluctuations in figure 6. Note that we only report the mean streamwise slip since we found the mean spanwise and wall-normal slips to be negligible. At $\delta _{\kern-1pt f}^+=50$ , the wall-scaled streamwise slip velocity is $\langle u_{\textit{slip}}\rangle /u_{\tau }=6.93$ . Increasing the filter size to $\delta _{\kern-1pt f}^+=300$ increases the wall-scaled streamwise slip to $\langle u_{\textit{slip}}\rangle /u_{\tau }=13.5$ . We also note that the streamwise slip shows a logarithmic trend with increasing $\delta _{\kern-1pt f}^+$ , as points in the logarithmic layer contribute greatly to the slip velocity at those larger filter sizes. Although much smaller than the mean, the streamwise slip fluctuations may be considered significant since $u'_{\textit{slip,rms}}/u_\tau \approx 1.5$ for all filter sizes. However, wall-normal and spanwise slip fluctuations are negligibly small since $v'_{\textit{slip,rms}}/u_\tau \approx 0.02$ and $w'_{\textit{slip,rms}}/u_\tau \approx 0.3$ , respectively. Slip fluctuations appear to plateau for $\delta _{\kern-1pt f}^+\geqslant 100$ , which indicates that these fluctuations may be caused by flow structures much larger than the filter sizes we considered.

Figure 7.

Instantaneous snapshots of slip velocity $u_{\textit{slip}}/u_{\tau }$ for $\delta _{\kern-1pt f}^+=50$ (a) and $300$ (b). The slip velocity shows the imprint of inner wall coherent flow structures, such as high- and low-speed streaks at $\delta _{\kern-1pt f}^+=50$ , and the imprint of outer layer structures $\delta _{\kern-1pt f}^+=300$ .

Far from being uniform or random, the slip velocity represents an imprint of the flow structures on the wall for relatively small filter sizes. At large filter sizes, most of the near-wall structures are filtered out, and the slip velocity field is representative of outer layer flow structures. We illustrate this in figure 7 where we show instantaneous snapshots of slip velocity at $\delta _{\kern-1pt f}^+=50$ and $300$ in a region of size $5000\times 2500$ in wall units. Clearly, the slip velocity field at $\delta _{\kern-1pt f}^+=50$ is dominated by fine-scale structures whereas the slip velocity field at $\delta _{\kern-1pt f}^+=300$ displays only large-scale structures. At $\delta _{\kern-1pt f}^+=50$ , the slip velocity field contains fine-scale structures that resemble high- and low-speed streaks characteristic of near-wall turbulence. In contrast, the slip velocity field at $\delta _{\kern-1pt f}^+=300$ displays higher-speed large-scale structures that represent the imprint of outer layer flow structures on the wall. The non-uniformities in this case are much lower than at $\delta _{\kern-1pt f}^+=50$ .

Figure 8.

Profiles of (a) streamwise, (b) wall-normal and (c) spanwise r.m.s. filtered velocity fluctuations for different filter sizes. The filtered profiles show the same trends as DNS profiles with better agreement near the centreline. The fluctuations decrease with increasing filter size. Line colours as in figure 5.

Figure 8 shows the r.m.s. velocity fluctuations from DNS and volume-filtered fields. With increasing filter size, the r.m.s. fluctuations in all three directions decrease since volume filtering suppresses fluctuations that occur at length scales smaller than the filter size. The volume-filtered r.m.s. profiles show the closest agreement with unfiltered profiles near the centreline. That is because fluctuations near the centreline are generated by large-scale motions that are little affected by filtering at $\delta _{\kern-1pt f}^+\leqslant 300$ . The discrepancy between filtered and unfiltered r.m.s. profiles is greatest close to the wall. In this region, near-wall fluctuations occur at smaller length scales. Filtering with $\delta _{\kern-1pt f}^+=50$ up to 300 removes a growing proportion of these fluctuations, which leads to lower r.m.s. values. All filtered profiles have non-zero values at the wall and decrease smoothly to zero at $\delta _{\kern-1pt f}/2$ into the wall as volume under the filter no longer contains fluid.

3.3. Volume-filtered stress balance

We now focus on analysing terms that appear in the volume-filtered momentum (2.22). Ensemble averaging this equation and considering the streamwise balance, we obtain the following balance:

(3.3) \begin{eqnarray} &&\alpha _{\kern-1pt f}\left ( \mu _{\kern-1pt f}\frac{\text{d}\left \langle \overline {u}_f\right \rangle }{\text{d}y} +\left \langle R_{\mu ,yx}\right \rangle -\rho _{\kern-1pt f}\left \langle \overline {v}'_f\overline {u}'_f\right \rangle -\rho _{\kern-1pt f}\left \langle \tau _{\textit{sfs},yx}\right \rangle \right ) \nonumber \\ && \quad=\tau _w\int _{-\infty }^y\left (-\frac {\alpha _{\kern-1pt f}}{h}+\mathcal{G}_1(y')+\mathcal{G}_1(y'-2h)\right )\text{d}y'. \end{eqnarray}

We provide a detailed derivation in Appendix A. The volume-filtered momentum balance (3.3) contains contributions from the resolved viscous stress ( $\mu _{\kern-1pt f}d\langle \overline {u}_f\rangle /\text{d}y$ ), residual viscous stress ( $\langle R_{\mu ,yx}\rangle$ ), Reynolds stress ( $-\rho _{\kern-1pt f}\langle \overline {v}'_f\overline {u}'_f\rangle$ ) and SFS stress ( $-\rho _{\kern-1pt f}\langle \mathsf{\tau }_{\textit{sfs},yx}\rangle$ ). The sum of all 4 contributions, which we refer to as the total stress, is equal to a function that depends on the wall-normal distance $y$ , wall shear stress $\tau _w$ , channel half-height $h$ , as well as the fluid volume fraction $\alpha _{\kern-1pt f}$ and filter $\mathcal{G}_1$ . The terms in $\mathcal{G}_1$ originate from the streamwise IB bodyforce $\langle F_{\textit{IB},x}\rangle$ . These terms are only active in a narrow band of width $\delta _{\kern-1pt f}$ around the walls (at $y=0$ and $y=2h$ ), but play an important role in setting the correct profile in these regions. Away from walls, the total stress approaches the linear function observed in DNS, i.e.

(3.4) \begin{eqnarray} &&\tau _w\int _{-\infty }^y\left (-\frac {\alpha _{\kern-1pt f}}{h}+\mathcal{G}_1(y')+\mathcal{G}_1(y'-2h)\right )\text{d}y'\approx \tau _w\left (1-\frac {y}{h}\right ) \nonumber \\ && \quad (\text{for } \delta _{\kern-1pt f}/2 \lesssim y \lesssim 2h-\delta _{\kern-1pt f}/2) . \end{eqnarray}

Close to walls, the total stress has a rounded profile that causes it to vanish inside the solid, i.e. at $y=-\delta _{\kern-1pt f}/2$ and $y=2h+\delta _{\kern-1pt f}/2$ .

Figure 9.

Mean stress balance for $\delta ^+_f=50$ (a, c) and $300$ (b,d) where (a,b) show profiles along the channel half-height and (c,d) show profiles in the near-wall region. The lines correspond to: normalised residual viscous stress $\alpha _{\kern-1pt f}\langle R_{\mu ,yx}\rangle /\tau _w$ ( ), normalised viscous stress $\alpha _{\kern-1pt f}\mu _{\kern-1pt f}(d\langle \overline {u}_f\rangle /\text{d}y)/\tau _w$ ( ), normalised Reynolds shear stress $-\alpha _{\kern-1pt f}\rho _{\kern-1pt f}\langle \overline {v}'_f\overline {u}'_f\rangle /\tau _w$ ( ), normalised SFS stress $-\alpha _{\kern-1pt f}\rho _{\kern-1pt f}\langle \mathsf{\tau }_{\textit{sfs},yx}\rangle /\tau _w$ ( ) and total filtered stress normalised by $\tau _w$ ( ). Total stress shows a linear profile away from the wall with a transiton region that scales with $\delta _{\kern-1pt f}$ near the wall. Increased filter size reduces the contributions of viscous and Reynolds stresses and increases the contributions of SFS and residual viscous stresses.

Figure 9 shows stress profiles at the finest ( $\delta _{\kern-1pt f}^+=50$ ) and coarsest ( $\delta _{\kern-1pt f}^+=300$ ) filter sizes tested. In both cases, the total stress has a narrow transition region at the wall and, otherwise, follows the linear profile from (3.4). As anticipated, this transition region is very narrow at $\delta _{\kern-1pt f}^+=50$ and wider at $\delta _{\kern-1pt f}^+=300$ . At $\delta _{\kern-1pt f}^+=50$ , the filtered Reynolds stress is comparable to the unfiltered one, while the combination of SFS, filtered viscous and residual viscous stresses is comparable to unfiltered viscous stress. At $\delta _{\kern-1pt f}^+=300$ , the filtered Reynolds and viscous stresses decrease significantly, which causes the SFS stress to adjust accordingly. At this larger filter size, the residual viscous stress is more significant than the filtered viscous stress although both make significantly reduced contributions compared with $\delta _{\kern-1pt f}^+=50$ .

Figure 10.

Profiles of (a) Reynolds, (b) SFS, (c) filtered viscous and (d) residual viscous stresses for filter sizes $\delta ^+_f=50$ - $300$ . Line colours as in figure 4.

Figure 10. Long description

The image contains four line graphs labeled Panel A, Panel B, Panel C, and Panel D. Each graph shows data profiles for different filter sizes. Panel A: This line graph displays the Reynolds stress profiles. The x-axis is labeled y/h and ranges from 0 to 1. The y-axis is labeled with a complex expression involving stress terms and ranges from 0 to 1. Multiple lines of different colors represent different filter sizes. Panel B: This line graph shows the SFS stress profiles. The x-axis is labeled y/h and ranges from 0 to 1. The y-axis is labeled with a complex expression involving stress terms and ranges from 0 to 0.6. Multiple lines of different colors represent different filter sizes. Panel C: This line graph presents the filtered viscous stress profiles. The x-axis is labeled y/h and ranges from 0 to 1. The y-axis is labeled with a complex expression involving stress terms and ranges from 0 to 0.3. An inset graph shows a zoomed-in view of the data for y+ ranging from -150 to 300. Multiple lines of different colors represent different filter sizes. Panel D: This line graph illustrates the residual viscous stress profiles. The x-axis is labeled y/h and ranges from 0 to 1. The y-axis is labeled with a complex expression involving stress terms and ranges from 0 to 0.3. An inset graph shows a zoomed-in view of the data for y+ ranging from -150 to 300. Multiple lines of different colors represent different filter sizes.

Lastly, to analyse the effect of filter size on the different filtered stresses, we plot in figure 10 the filtered viscous, residual viscous, Reynolds and SFS stress profiles for all filter sizes tested. The filtered Reynolds stress has the shape expected from a classical channel flow where it dominates in the free stream but is negligible in the near-wall region relative to the other stress contributions. Increasing the filter size lowers the filtered Reynolds stress and shifts its peak location towards the centreline. The SFS stress has a form that is reminiscent of the unfiltered viscous stress, where it is largest close to the wall and vanishes towards the centreline. Increasing the filter size causes this term to gain in magnitude as its normalised peak grows from 0.24 at $\delta _{\kern-1pt f}^+=50$ to 0.57 at $\delta _{\kern-1pt f}^+=300$ . The filtered viscous stress and residual viscous stress are significant only near the wall. Although the latter is larger across all filter sizes tested, both terms have comparable normalised magnitudes. With increasing filter size, both the residual and filtered viscous stresses drop in magnitude. Note that, while the filtered viscous stress has a long vanishing tail that extends towards the channel centre, the residual viscous stress has compact support centred on the region from $-\delta _{\kern-1pt f}/2$ to $\delta _{\kern-1pt f}/2$ due to our choice of compact filter.

4.1. Algebraic slip velocity model

A simple model for the mean wall-slip velocity $\langle u_{\textit{slip}}\rangle$ can be formulated by volume-filtering an assumed velocity profile. In this study, we use the VD model (Van Driest Reference Van Driest1956), which gives the mean streamwise velocity profile as

(4.1) \begin{eqnarray} \frac {\left \langle u^{\textit{VD}}\right \rangle (y)}{u_\tau }=\int _0^{y/\delta _\nu }\frac {2}{1+\sqrt {1+4(\kappa y'(1-\exp (-y'/A^+)))^2}}\text{d}y', \end{eqnarray}

where $\kappa = 0.41$ , $A^+=26$ and $\delta _\nu =\nu /u_\tau$ . Applying the volume filter to this profile, we obtain the following slip model:

(4.2) \begin{align} \frac {\left \langle u_{\textit{slip}}^{\textit{VD}}\right \rangle (\delta _{\kern-1pt f})}{u_\tau }=\frac {1}{\alpha _{f,w}}\int _0^{\delta _{\kern-1pt f}/2}\int _0^{y/\delta _\nu }\frac {2}{1+\sqrt {1+4(\kappa y'(1-\exp (-y'/A^+)))^2}}\mathcal{G}_1(y)\text{d}y'\text{d}y. \end{align}

Figure 11.

(a) Normalised streamwise slip velocity computed by filtering DNS data, numerical evaluation of the VD slip (4.2) and the VD slip approximation (4.5), and (b) relative error for the VD slip approximation at different filter sizes. The VD slip model shows excellent agreement with the filtered DNS data. The VD approximation (4.5) yields less than $1\,\%$ error for $\delta _{\kern-1pt f}^+\geq 100$ but diverges for $\delta _{\kern-1pt f}^+\rightarrow 0$ .

The VD slip model (4.2) is a very good model for channel flow turbulence and similarly simple flow configurations. To show this, we report in figure 11(a) a comparison of the mean slip velocity obtained by directly volume filtering the turbulent channel flow DNS $\langle u_{\textit{slip}}\rangle$ and the slip derived from the VD model $\langle u_{\textit{slip}}^{\textit{VD}}\rangle$ . Clearly, there is excellent agreement between the two for filter sizes $\delta _{\kern-1pt f}^+=50$ to 300.

Despite reproducing the streamwise slip very well, the VD slip model has a major inconvenience, which is that the nested double integrals in (4.2) cannot be computed analytically. While numerical integration is acceptable in a priori analyses, it is too costly and impractical in forward LES runs.

To alleviate this issue, we propose an approximation of (4.2) that is much simpler to evaluate, yet provides accurate results for filter sizes $\delta _{\kern-1pt f}^+\geqslant 50$ . First, we note that, for very large filter sizes, the VD slip $\langle u_{\textit{slip}}^{\textit{VD}}\rangle$ tends towards a logarithmic profile, i.e.

(4.3) \begin{align} \left \langle u_{\textit{slip}}^{\textit{VD}}\right \rangle /u_\tau \approx \frac {1}{\alpha _{f,w}}\frac {1}{2\kappa }\log (\delta _{\kern-1pt f}^+/2) \quad \text{for}\;\delta _{\kern-1pt f}^+\gg 1. \end{align}

Thus, we can express the VD slip in terms of the logarithmic term above and a remainder function $f(\delta _{\kern-1pt f}^+)$ such that

(4.4) \begin{align} \left \langle u_{\textit{slip}}^{\textit{VD}}\right \rangle /u_\tau = \frac {1}{\alpha _{f,w}}\left (\frac {1}{2\kappa }\log (\delta _{\kern-1pt f}^+/2) +f(\delta _{\kern-1pt f}^+)\right )\!. \end{align}

We determine an approximation for $f$ using a power fit. Injecting it in (4.4), we obtain the following approximation for the VD slip:

(4.5) \begin{align} \left \langle u_{\textit{slip}}^{\textit{VD,approx}}\right \rangle /u_\tau = \frac {1}{\alpha _{f,w}}\left (\frac {1}{2\kappa }\log (\delta _{\kern-1pt f}^+/2) + 0.972 - 72.0\left (1/\delta ^+_f\right )^{0.9523} \right ) . \end{align}

Figure 11(b) shows that the relative error between the VD slip approximation $\langle u_{\textit{slip}}^{\textit{VD,approx}}\rangle$ from (4.5) and the exact VD slip $\langle u_{\textit{slip}}^{\textit{VD}}\rangle$ from (4.2) is less than 1 % for filter sizes $\delta _{\kern-1pt f}^+\geqslant 100$ but diverges for $\delta _{\kern-1pt f}^+\rightarrow 0$ .

4.2. Slip-length model

This second strategy is a variation on the slip model proposed by Bose & Moin (Reference Bose and Moin2014) and later expanded on by Bae et al. (Reference Bae, Lozano-Durán, Bose and Moin2019). Bose & Moin (Reference Bose and Moin2014) suggested that a wall model can be formulated by finding a so-called slip length, $\ell _x$ , such that the following relationship holds for the streamwise slip:

(4.6) \begin{eqnarray} \left \langle u_{\textit{slip}}\right \rangle = \ell _x \left .\left \langle \frac {\partial \overline {u}_f}{\partial y}\right \rangle \right |_w . \end{eqnarray}

They arrived at this conclusion by matching leading-order terms in Taylor series expansions of the wall slip and filtered velocity wall gradient. A detailed description of a dynamic method of determining this slip length is given in Bae et al. (Reference Bae, Lozano-Durán, Bose and Moin2019), and will not be discussed in this paper. Although this approach has been shown to yield good results in cases with simple walls, namely, turbulent channel flows, the approach of Bose & Moin (Reference Bose and Moin2014) suffers from the commutation errors discussed in § 2, and does not account for non-planar, moving or deforming boundaries.

Here, we derive a slip-length model using different arguments from Bose & Moin (Reference Bose and Moin2014) and in a way that is adapted to the volume-filtering framework. Applying identity (2.11) with $\boldsymbol{\phi }=\boldsymbol{u}_{\kern-1pt f}$ and evaluating at the wall, we get

(4.7) \begin{eqnarray} \left .\overline {\boldsymbol{\nabla }\boldsymbol{u}_{\kern-1pt f}}\right |_w - \left .\boldsymbol{\nabla }\overline {\boldsymbol{u}}_{\kern-1pt f}\right |_w = \frac {1}{\alpha _{f,w}} \left (\left .\boldsymbol{\nabla }\alpha _{\kern-1pt f}\right |_w \right )\boldsymbol{u}_{\textit{slip}}. \end{eqnarray}

It is noteworthy that the tensorial identity (2.11) is general since it applies to planar, non-planar, static and moving walls alike, provided that the filter size $\delta _{\kern-1pt f}$ is small enough to resolve the local curvature and any variation of the local wall velocity. Next, we take the dot product of the wall-normal unit vector with (2.11), taking care to multiply from the left. Rearranging the equation, we obtain a relationship for the slip velocity in the form

(4.8) \begin{align} \boldsymbol{u}_{\textit{slip}} = \frac {\alpha _{f,w}}{\left .\partial \alpha _{\kern-1pt f}/\partial n\right |_w}\left ( \left .\overline {\frac {\partial \boldsymbol{u}_{\kern-1pt f}}{\partial n}}\right |_w - \left .\frac {\partial \overline {\boldsymbol{u}}_{\kern-1pt f}}{\partial n}\right |_w \right )\!. \end{align}

To formulate our slip model, we introduce the non-dimensional coefficients $\boldsymbol{\lambda }=(\lambda _{n},\lambda _{t_1},\lambda _{t_2})$ , such that

(4.9) \begin{eqnarray} \lambda _{n} = \frac {\left \langle \left .\overline {\partial u_{f,n}/\partial n}\right |_w\right \rangle }{\left \langle \left .\partial \overline {u}_{f,n}/\partial n\right |_w\right \rangle }; \quad \lambda _{t_1} = \frac {\left \langle \left .\overline {\partial u_{f,t_1}/\partial n}\right |_w\right \rangle }{\left \langle \left .\partial \overline {u}_{f,t_1}/\partial n\right |_w\right \rangle }; \quad \lambda _{t_2} = \frac {\left \langle \left .\overline {\partial u_{f,t_2}/\partial n}\right |_w\right \rangle }{\left \langle \left .\partial \overline {u}_{f,t_2}/\partial n\right |_w\right \rangle }, \end{eqnarray}

where the ‘ $n$ ’ notation signifies components in the wall-normal direction, the ‘ $t_1$ ’ and ‘ $t_2$ ’ notations signify components in the wall-parallel directions. Substituting (4.9) into ensemble-averaged (4.8) gives a slip formulation similar to the one in Bose & Moin (Reference Bose and Moin2014), i.e.

(4.10) \begin{align} \left \langle u_{\textit{slip},n} \right \rangle =\ell _{n} \left \langle \left .\frac {\partial \overline {u}_{f,n}}{\partial n}\right |_w\right \rangle \!, \\[-28pt] \nonumber \end{align}

(4.11) \begin{align} \left \langle u_{\textit{slip},t_1} \right \rangle =\ell _{t_1} \left \langle \left .\frac {\partial \overline {u}_{f,t_1}}{\partial n}\right |_w\right \rangle \!, \\[-28pt] \nonumber \end{align}

(4.12) \begin{align} \left \langle u_{\textit{slip},t_2} \right \rangle =\ell _{t_2} \left \langle \left .\frac {\partial \overline {u}_{f,t_2}}{\partial n}\right |_w\right \rangle \!, \\[0pt] \nonumber \end{align}

and where the slip lengths $\boldsymbol{\ell }=(\ell _{n},\ell _{t_1},\ell _{t_2})$ are

(4.13) \begin{align} \boldsymbol{\ell } = \frac {\alpha _{f,w}}{\left .\partial \alpha _{\kern-1pt f}/\partial n\right |_w}(\boldsymbol{\lambda }-1). \end{align}

Non-dimensionalising by wall units, we obtain the wall-scaled slip velocities and slip lengths

(4.14) \begin{align} \frac {\left \langle u_{\textit{slip},n} \right \rangle }{u_\tau }=\ell _{n}^+ \left \langle \left .\frac {\partial \overline {u}_{f,n}^+}{\partial n^+}\right |_w\right \rangle \!, \\[-28pt] \nonumber \end{align}

(4.15) \begin{align} \frac {\left \langle u_{\textit{slip},t_1} \right \rangle }{u_\tau }=\ell _{t_1}^+ \left \langle \left .\frac {\partial \overline {u}_{f,t_1}^+}{\partial n^+}\right |_w\right \rangle \!, \\[-28pt] \nonumber \end{align}

(4.16) \begin{align} \frac {\left \langle u_{\textit{slip},t_2} \right \rangle }{u_\tau }=\ell _{t_2}^+ \left \langle \left .\frac {\partial \overline {u}_{f,t_2}^+}{\partial n^+}\right |_w\right \rangle \!, \\[-28pt] \nonumber \end{align}

(4.17) \begin{align} \boldsymbol{\ell }^+ =\frac {\alpha _{f,w}}{\left .\partial \alpha _{\kern-1pt f}/\partial n^+\right |_w}(\boldsymbol{\lambda }-1). \\[0pt] \nonumber \end{align}

Equations (4.14)–(4.17) represent the wall-scaled slip velocities and slip lengths in general wall coordinates for a geometry that has slight curvature compared with the filter size. For the channel flow configuration presented here, the ‘ $t_1$ ’ direction corresponds to the streamwise direction $x$ , the ‘ $n$ ’ direction corresponds to the wall-normal direction $y$ and the ‘ $t_2$ ’ direction corresponds to the spanwise direction $z$ . Thus, the wall-scaled slip velocities and slip lengths have the form

(4.18) \begin{align} \frac {\left \langle u_{\textit{slip}} \right \rangle }{u_\tau } = \ell _x^+ \left \langle \left .\frac {\partial \overline {u}_{\kern-1pt f}^+}{\partial y^+}\right |_w\right \rangle \!, \\[-28pt] \nonumber \end{align}

(4.19) \begin{align} \frac {\left \langle v_{\textit{slip}} \right \rangle }{u_\tau } = \ell _y^+ \left \langle \left .\frac {\partial \overline {v}_{\kern-1pt f}^+}{\partial y^+}\right |_w\right \rangle \!, \\[-28pt] \nonumber \end{align}

(4.20) \begin{align} \frac {\left \langle w_{\textit{slip}} \right \rangle }{u_\tau } = \ell _z^+ \left \langle \left .\frac {\partial \overline {w}_{\kern-1pt f}^+}{\partial y^+}\right |_w\right \rangle \!, \\[-28pt] \nonumber \end{align}

(4.21) \begin{align} \boldsymbol{\ell }^+ = \frac {\alpha _{f,w}}{\mathcal{G}_1^+(0)}(\boldsymbol{\lambda }-1), \\[0pt] \nonumber \end{align}

where we also use the relationship $ (\partial \alpha _{\kern-1pt f}/\partial y^+ )|_w=\mathcal{G}_1^+(0)$ for planar walls. Correspondingly, the non-dimensional coefficients $\boldsymbol{\lambda }=(\lambda _x,\lambda _y,\lambda _z)$ are

(4.22) \begin{align} \lambda _x = \frac {\left \langle \left .\overline {\partial u_{\kern-1pt f}/\partial y}\right |_w\right \rangle }{\left \langle \left .\partial \overline {u}_f/\partial y\right |_w\right \rangle }; \quad \lambda _y = \frac {\left \langle \left .\overline {\partial v_f/\partial y}\right |_w\right \rangle }{\left \langle \left .\partial \overline {v}_f/\partial y\right |_w\right \rangle }; \quad \lambda _z = \frac {\left \langle \left .\overline {\partial w_f/\partial y}\right |_w\right \rangle }{\left \langle \left .\partial \overline {w}_f/\partial y\right |_w\right \rangle }. \end{align}

Equations (4.18)–(4.21) relate the slip velocities to the wall-normal gradients, volume fraction at the wall $\alpha _{f,w}$ , wall-scaled filter size through $\mathcal{G}_1^+(0)\propto 1/\delta _{\kern-1pt f}^+$ and non-dimensional coefficients $(\lambda _x,\lambda _y,\lambda _z)$ . Among these dependencies, only $(\lambda _x,\lambda _y,\lambda _z)$ are unclosed. Therefore, closing these coefficients with an appropriately chosen model also closes the slip lengths $(\ell ^+_x,\ell ^+_y,\ell ^+_z)$ .

It is clear that $\boldsymbol{\lambda }$ must depend on the filter size $\delta _{\kern-1pt f}$ , and possibly on the friction Reynolds number $ \textit{Re}_\tau$ . Thus, we can write formally

(4.23) \begin{align} \boldsymbol{\lambda }=\boldsymbol{\lambda }(\delta _{\kern-1pt f}^+, Re_\tau ,\ldots ) . \end{align}

In the limit of vanishingly small filter sizes

(4.24) \begin{align} (\lambda _x,\lambda _y,\lambda _z) \rightarrow (1,1,1) \quad \text{for}\; \delta _{\kern-1pt f}\rightarrow 0 , \end{align}

since both nominators and denominators in equations (4.22) become equal to one another. This ensures that the slip lengths also vanish ( $\boldsymbol{\ell }\rightarrow 0$ ) when $\delta _{\kern-1pt f}\rightarrow 0$ . Given that the data set we use in this study corresponds to a single $ \textit{Re}_\tau$ , we are unable to tease out the exact functional dependence of $\boldsymbol{\lambda }$ on $ \textit{Re}_\tau$ , but expect $ \textit{Re}_\tau$ dependence to be second order to $\delta _{\kern-1pt f}^+$ dependence. There can also be additional dependencies in the case of non-planar, rough, moving or deforming walls, although we do not investigate these effects in this study.

Although a universal model for $(\lambda _x,\lambda _y,\lambda _z)$ may be difficult to formulate, we propose next closures that apply to the turbulent channel flow and similarly simple wall-bounded flows.

The a priori analysis in § 3 shows that the wall-normal and spanwise slip velocities are negligible in a turbulent channel flow. Thus, the slip lengths $\ell _y$ and $\ell _z$ are negligible. A model that reproduces this observation consists of simply taking $\lambda _y$ and $\lambda _z$ constant and unitary i.e.

(4.25) \begin{align} \lambda _y = 1 \quad \text{and}\quad \lambda _z = 1. \end{align}

Figure 12.

Joint PDFs of $\overline {\partial u_{\kern-1pt f}^+/\partial y^+}|_w$ and $\partial \overline {u}_{\kern-1pt f}^+/\partial y^+|_w$ normalised by peak values at $\delta _{\kern-1pt f}^+=50{-}300$ (left to right, top to bottom). The oblong PDF distribution shows clear correlation between $\overline {\partial u_{\kern-1pt f}^+/\partial y^+}|_w$ and $\partial \overline {u}_{\kern-1pt f}^+/\partial y^+|_w$ at all filter sizes.

Figure 13.

Variation of (a) the non-dimensional coefficient $\lambda$ and (b) normalised slip length $\ell ^+$ as a function of $\delta _{\kern-1pt f}^+$ computed from filtered DNS data along with accompanying power-law fits. Power-law fits for $\lambda _x$ and $\ell ^+_x$ show good agreement for all filter sizes tested with $\lambda _x$ scaling as $\delta _{\kern-1pt f}^{+ \, 0.5385}$ and $\ell ^+_x$ scaling as $\delta _{\kern-1pt f}^{+\,1.5385}$ at large filter sizes.

To elucidate the functional dependence of the streamwise $\lambda _x$ on $\delta _{\kern-1pt f}^+$ , we compute and report in figure 12 joint probability density functions (PDFs) of $ \overline {\partial u_{\kern-1pt f}^+/\partial y^+} |_w$ and $ \partial \overline {u}_{\kern-1pt f}^+/\partial y^+ |_w$ . At $\delta _{\kern-1pt f}^+=50$ , the joint PDF isocontours are elongated and titled upwards, which shows a clear correlation between $ \overline {\partial u_{\kern-1pt f}^+/\partial y^+} |_w$ and $ \partial \overline {u}_{\kern-1pt f}^+/\partial y^+ |_w$ . The peak value is located at $\approx (0.2,0.6)$ . With increasing filter size, the peak value shifts towards left and downwards, i.e. towards lower $ \overline {\partial u_{\kern-1pt f}^+/\partial y^+} |_w$ and $ \partial \overline {u}_{\kern-1pt f}^+/\partial y^+ |_w$ values. This is because increasing filter size suppresses further the small scales responsible for large velocity gradients, as explained in § 3. The joint PDF iso-contours are also elongated at an angle that gets steeper with increasing filter size.

From the joint PDFs, we extract the lines of best fit shown in black in figure 12. The slope of these lines gives the non-dimensional coefficient $\lambda _x$ . From there, we determine $\ell ^+_x$ with (4.21) using $\alpha _{f,w}=1/2$ , and $\mathcal{G}_1^+(0)=\pi /(2\delta _{\kern-1pt f}^+)$ for the cosine filter. Figure 13(a) shows the resulting $\lambda _x$ values for different filter sizes. As anticipated, $\lambda _x$ is equal to 1 at $\delta _{\kern-1pt f}=0$ and increases with $\delta _{\kern-1pt f}$ . A power fit gives a good approximation of the functional dependence of $\lambda _x$ on $\delta _{\kern-1pt f}$

(4.26) \begin{align} \lambda _x = 1 + 0.251 \delta _{\kern-1pt f}^{+\,0.5385} . \end{align}

Injecting this form in (4.21) gives the following slip-length model:

(4.27) \begin{align} \ell ^+_x = 0.0798 \delta _{\kern-1pt f}^{+\,1.5385} \end{align}

for the cosine filter. Figure 13(b) shows that this model captures indeed well the slip length that we obtain directly by volume filtering the DNS data.