Predicting internal boundary layer growth following a roughness change in thermally neutral and stable boundary layers

Abstract

This study uses the diffusion analogy (Miyake, Sci. Rep., 5R-6, 1965, Univ. of Washington, Seattle, USA) to predict the full growth behaviour of internal boundary layers (IBLs) induced by a roughness change for neutrally – and especially stably – stratified boundary layers with finite thickness. The physics of the diffusion analogy shows that the streamwise variation of the IBL thickness is dictated by $\sigma _w/U$ at the interface, where $\sigma _w$ and $U$ represent wall-normal Reynolds stress and mean streamwise velocity, respectively. The existing variants of the model, summarised by Savelyev & Taylor (2005, Boundary-Layer Meteorol., vol. 115, pp. 1–25), are tailored to IBLs confined within the constant shear stress layer. To extend the applicability of the model to the outer region, we investigate the relation between $\sigma _w/U$ and $U/U_\infty$ in the outer region across varying stratification, where $U_\infty$ is the free-stream velocity. Our analysis reveals that wind tunnel data from a number of facilities collapse onto a master curve when $\sigma _w/U$ is premultiplied by a height-independent parameter, which is a function of the ratio of Monin–Obukhov length to the boundary layer thickness. The scaled $\sigma _w/U$ decreases inversely with $U/U_\infty$ in the surface layer, transitioning to a linear decrease as $U/U_\infty$ increases. The new model, which integrates these findings, along with the effects of streamline displacement and acceleration, captures the complete characteristics of IBLs as they develop within turbulent boundary layers of finite thickness.

1. Introduction

Changes in the aerodynamic or thermal properties of surfaces beneath the atmospheric boundary layer generate an interface capping an internal boundary layer (IBL), where the energy and momentum fluxes reflect a blend of characteristics from the surfaces before and after the change. Examples of the transition include flows from rural to urban regions, from sea to land, and from sea to ice cap (and vice versa), which influence the local pollutant dispersion and meteorology (Baklanov et al. 2011). Studying the IBL is beneficial for applications such as weather prediction, pollutant management and wind energy optimisation. Consequently, the accurate prediction of the IBL growth curve has been a longstanding research focus (Garratt 1990; Bou-Zeid et al. 2020), especially when the boundary layer is coupled with multiple physical processes and on different length scales.

A paradigm model for studying kinetic IBLs emphasises an abrupt change in surface roughness along the streamwise direction, where the surface downstream of the change has the roughness length $z_{02}$ , and upstream of the change $z_{01}$ . Predicting the development of the adjusted layer, i.e. the variation of its thickness $\delta _i$ along the streamwise fetch $x$ , begins with an important simplification, namely that this layer is embedded within the constant shear stress region.

Miyake (1965) proposed a passive diffusion analogy, which links the IBL growth to the vertical dispersion of a passive plume. The IBL’s growth rate ${{\rm d}\delta _i}/{{\rm d}t}$ is prescribed by $\sigma _w$ , i.e. the standard deviation of the wall-normal velocity $w$ at the interface. Using the chain derivative rule ${\rm d}x=U\,{\rm d}t$ leads to $U({{\rm d}\delta _i}/{{\rm d}x})=A \sigma _w$ , where $\sigma _w$ is a height-independent value as the IBL lies within the constant shear stress layer. The constant $A$ is model-dependent, though Savelyev & Taylor (2001) revealed that it increases with $M\equiv \ln ({z_{02}}/{z_{01}})$ by studying a large number of datasets. The relationship between $A$ and $M$ is associated with the effect of streamlines displacement across the step change in roughness, which was accounted for by incorporating the mean wall-normal velocity into the model by Savelyev & Taylor (2005). In the same work, they extended the model to diabatic conditions by integrating the Monin–Obukhov similarity (MOST) (Monin & Obukhov 1954) to predict the development of the IBL across a change in the Monin–Obukhov length.

The simplified scenario considered in the aforementioned model becomes less effective when the outer edge of the IBL exceeds the surface layer height due to the deviation of the mean streamwise velocity from the logarithmic law (MOST) for neutral (stable) flows, and the significant variation of $\sigma _w$ with height. Conventionally, the height–fetch ratio estimated by these small fetch models was employed to predict the complete evolution of the IBL (i.e. through the whole boundary layer) assuming the absence of Coriolis force effects (Garratt 1990). However, in wind tunnel studies, the overprediction of the IBL formulae from the physical models was remarkable, with surface changes from rough to smooth (Li et al. 2022) and rough to rougher in neutral conditions (Gul & Ganapathisubramani 2022) and in stable conditions (Ding et al. 2023). For neutral stratification, Li et al. (2022) predicted a shallower IBL in the outer region by developing the physical model initially proposed by Elliott (1958) for the logarithmic layer in the context of a finite-thickness boundary layer. They consider the modification of streamwise momentum within the IBL to be dictated by the decrease in the shear stress across this region. Accounting for the decay of the shear stress with height, the wake function, and the thickening process of the boundary layer collectively slow the IBL growth in the outer region.

However, to the authors’ knowledge, there is no existing physical model that captures the complete growth of the IBL (irrespective of the relative depth of the IBL compared to the surface layer) for stable thermal stratification. Whether or not the diffusion analogy can accurately capture the characteristics of the IBL in the outer region, especially under stable conditions, has yet to be decided. To address this question, we investigate the impact of stable stratification on $\sigma _w/U$ and $U$ within the boundary layer using data measured in different wind tunnels. Subsequently, we develop a modified diffusion model that can accommodate any IBL depth.

2. Cases studied

We focus on surface roughness changes from rough ( $z_{01}$ ) to rougher ( $z_{02}\gt z_{01}$ ) using the published data in Ding et al. (2024, 2024). Only the essential information relevant to this paper is included below; additional details can be found in the referenced papers. Streamwise velocity ( $\tilde {u}=U+u$ ) and vertical velocity ( $\tilde {w}=W+w$ ) were measured using laser Doppler anemometry. Here, $(U, W)$ denotes the mean value, and $(u, w)$ denotes the fluctuations. Profiles along the wall-normal direction $z$ were measured in the central plane of the wind tunnel, where the origin of the coordinates is set at the roughness change on the floor. The aerodynamic parameters of the incoming flow were measured at $0.76$ m upstream of the step change in roughness.

Four cases, comprising two neutrally and two stably stratified boundary layers, were investigated. The characteristics of the incoming flows and the roughness length change are summarised in table 1. In the table, the Reynolds number is $Re=U_\infty \delta /\nu$ , where $\nu$ denotes the kinematic viscosity, $U_\infty$ denotes the free-stream velocity, and $\delta$ denotes the boundary layer thickness determined by the mean streamwise velocity reaching 99 % of the free-stream velocity $U_\infty$ . The bulk Richardson number is $Ri_b=({g(\Theta _\delta -\Theta _0)\delta })/{\Theta _0 U_\delta ^2}$ , where $\Theta _0$ denotes the absolute surface temperature, $g$ is gravitational acceleration, and $\Theta _\delta$ ( $U_\delta$ ) denotes the mean temperature (streamwise velocity) at the top of the boundary layer. The surface Monin–Obukhov length is $L_0=u_*^2\Theta _0/(g\kappa \theta _*)$ and is a measure of the height at which stability leads to significant reductions in vertical turbulence. Here, $u_*$ ( $\theta_*$ ) represents the friction velocity (temperature).

Table 1. Dimensionless parameters for the cases studied: $Re,\ Ri_b,\ z_{01}/\delta$ and $L_{01}/\delta$ are for the incoming flow, while $z_{02}/\delta$ and $L_{02}/\delta$ for the downstream flow.

For both neutral and stable cases, the IBL depth was determined from the merging point of the local wall-normal profile of $\sigma ^2_u(z)$ with its counterpart upstream of the roughness change. This methodology is consistent with identifying the ‘knee’ point in the mean streamwise velocity profile (Li et al. 2022; Ding et al. 2023). Here, $\sigma _u$ denotes the standard deviation of the streamwise velocity. This methodology, and the subsequent analysis in § 3, relies on the assumption of spanwise homogeneity, which was demonstrated in the region above the roughness sublayer (Ding et al. 2024).

Variation in the degree of stratification, conventionally induced by the change in wall temperature or heat flux, can generate a thermal IBL that caps a region with modified mean temperature or heat flux (Garratt 1990). The cases discussed here do not present any discontinuity in the surface thermal condition; across the roughness, change $L_0/\delta$ varies from 1.0 to 1.9 for case S1, and from 0.6 to 1.2 for case S2. This variation in the stratification is not significant as the thermal IBL passively follows the kinetic IBL (Ding et al. 2024). Thus we focus on the latter in the following analysis.

3. Results and discussion

3.1. Predicting wall-normal turbulent intensity profiles in stable flows

To extend the diffusion analogy to the outer region, we begin with analysis of $\sigma _w/U$ . Figure 1(a) shows $\sigma _w/U$ as a function of $U/U_\infty$ (the lower branch) for the logarithmic regions and beyond, which are the target predictive regions. The curves of $\sigma _w/U$ in stable cases were scaled by a height-independent parameter $\sigma _L$ to collapse results onto the neutral ones. The value of $\sigma _L$ was determined by minimising the difference between $\sigma _L\sigma _w/U$ for the stable case and $\sigma _w/U$ for neutral cases in the region of interest. The parameter, $\sigma _L$ , measures the suppression of the wall-normal component of normal stress due to stable stratification. As shown in figure 1(c), $\sigma _L$ decreases monotonically as $L_0/\delta$ increases. Moreover, data from different datasets fall onto a unique function in the form $\sigma _L=0.9256e^{-L_0/\delta }+1$ . This exponential decay is demonstrated by the linear function in the semi-log plot in figure 1(d). However, the relation between $\sigma _L$ and $Ri_b$ is sensitive to the system investigated, as shown in figure 1(b).

Figure 1(a) demonstrates good data collapse as $\sigma _L\sigma _w/U$ varies with $U/U_\infty$ consistently in all cases. In the logarithmic layer, the attached-eddy hypothesis proposed by Townsend (1976, pp. 144–145) prescribes that $\sigma _w$ is independent of $z$ . This height invariance is a foundational assumption in the diffusion analogue model, where $\sigma _w$ is typically taken as 1.25 $u_*$ within the region extending from $z=z_{02}$ (where $U/U_\infty \sim 0.1M$ ) to the upper bound of the logarithmic layer. This assumption leads to $\sigma _w/U \sim {U}^{-1}$ . Preserving this relationship within (3.1) below ensures that the model in § 3.3 approaches the classical one as $U \to 0$ for an infinitely small $M$ , since all terms of order $\sim o(U^n)$ (with $n\geqslant 1$ ) vanish in the aforementioned limit. In the wake region ( $0.75 \leqslant U/U_{\infty }$ ), $\sigma _L\sigma _w/U$ decreases approximately as a linear function of $U/U_{\infty }$ . These trends are consistent with previous observations of boundary layers on smooth surfaces under neutral (Flack, Schultz & Shapiro 2005) and stable (Williams et al. 2017) conditions. To delineate the complete trend, we propose a composite function

(3.1) \begin{equation} \frac {\sigma _L\sigma _{w}}{U}=a_0 {\left (\frac {U}{U_\infty }\right )}^{-1}-{a_1}\frac {U}{U_\infty }, \end{equation}

where $a_0=0.0693$ and $a_1=0.0476$ are fitting parameters for the master curve shown in figure 1(a). We expect this curve to be valid for different rough walls provided that they obey Townsend’s similarity hypothesis (Flack et al. 2005, 2007).

Figure 1(a) also demonstrates $\sigma _L\sigma _u/U$ as a function of $U/U_\infty$ . Here, the collapse onto a master linear curve is poorer than seen for the wall-normal components. Their relation within $0.7\lt U/U_\infty \lt 1$ follows the empirical function, i.e. $\sigma _u/U=0.286-0.255U/U_{\infty }$ , identified by Alfredsson, Segalini & Örlü (2011) for neutral boundary layers over smooth surfaces. The curve becomes steeper with increasing roughness (Castro 2013). This observation suggests that the empirical function could be reliably employed to $\sigma _L\sigma _u/U$ in stable boundary layers over smooth or transitionally rough surfaces.

Figure 1. (a) Plots of $\sigma _L \sigma _w U^{-1}$ (lower branch, empty symbols) and $\sigma _L \sigma _u U^{-1}$ (upper branch, solid symbols) as functions of $U/U_\infty$ . The red dotted line indicates $( U/U_{\infty} )^{-1}$ . The red solid curve is the fit by (3.1). The red dashed line is the relation proposed by Alfredsson et al. (2011). Plots of $\sigma _L$ as a function of (b) bulk Richardson number $Ri_b$ and (c) $L_0/\delta$ . (d) Plot of $\sigma _L-1$ as a function of $L_0/\delta$ . In (b,c,d), solid circles (squares) are for case S1 (S2) from Ding et al. (2024), and empty symbols are from Williams et al. (2017) with the notation in (a).

3.2. Empirically modelling mean profiles of streamwise velocity

Figure 2(a) demonstrates that the effect of stratification on the mean streamwise velocity profiles is restricted within $z_c/\delta \lt 0.25$ as $U/U_\infty$ therein is sensitive to the degree of stratification. We define $z_c$ as a precise measure of the thickness of the layer where the mean streamwise velocity is significantly influenced by stratification. In the region beyond $z_c$ , $U/U_\infty$ is barely altered by the stratification (variations within $4\,\%$ ), as shown in figure 2(a). This suggests that the wake function in adiabatic conditions can be used extensively for stably stratified boundary layers. Therefore, we employ MOST in the region $z\lt z_c$ (under the assumption that the thickness of the surface layer with a constant flux equals $z_c$ ), thus the stability function $\phi _m$ , which links the gradient of mean streamwise velocity to the surface momentum flux in the form ${\phi _m}=({\kappa z}/{u_*}) ({{\rm d} U}/{{\rm d} z})$ , was modified to $\phi _m(z/L_0)=1+({\beta _m z}/{(2L_0)})\,\textrm {erfc} ( ({z-z_c})/{s} )$ . Integrating ${\rm d}U/{\rm d}z$ from $z_0$ and adopting the wake function (Jones, Marusic & Perry 2001) constructs the complete vertical profile of $U/U_\infty$ assuming a negligible zero-plane displacement (this assumption restricts the applicability of this approach in large urban settings, where the zero-plane displacement can be significant, but it is reasonable in many atmospheric flows):

(3.2) \begin{equation} U_{{n}} \equiv \frac {U}{U_{\infty }}=\sqrt {\frac {C_f}{2}}\frac {1}{\kappa } \left \{\ln {\frac {z}{z_{01}}}+\frac {\beta _m}{L_0}\Phi (z,z_c,s)+\Pi \left [ 2 \left ( \frac {z}{\delta _c} \right )^2 \left (3-\frac {2z}{\delta _c}\right ) \right ]-\frac {1}{3}\left (\frac {z}{\delta _c}\right )^2\right \}_{_{_{}}}\!, \end{equation}

where $\Phi =\int _{z_{01}}^z (1/2)\,\textrm{erfc}(({l-z_c})/{s})\, {\rm d}l$ . The skin-friction coefficient is $C_f=2\rho _w u_*^{2}/ (\rho _\infty U_\infty ^{2})$ , with $\rho _w$ ( $\rho_\infty$ ) the air density at the wall (in the free stream). Here, $\beta _m=8$ for the studied flows (Hancock & Hayden 2018), $\Pi$ represents the wake factor, and $s$ measures the width of the transition region between two layers with or without significant influences of stratification, being approximately $0.1z_c$ for stable cases. The real boundary layer thickness $\delta _c$ approximates $1.2\delta$ for cases studied. Figure 2(a) shows a range of measured data against (3.2).

Marusic et al. (2015) found that $\Pi$ is independent of the streamwise location for fully developed neutral boundary layers. Herein we use the terminology ‘fully developed’ to refer to boundary layers that are fully adjusted to the underlying surface and whose characteristics are slowly varying in the streamwise direction. Hancock & Hayden (2018) noted the invariance of wall-normal profiles of mean streamwise velocity with streamwise locations for fully developed stable boundary layers. Thus $U(z)/U_\infty$ should be dictated by $(\Pi ,z_0/\delta , C_f, L_0/\delta , z_c/\delta)$ . Substituting $z=\delta$ into (3.2) gives the relation between $C_f$ and $(\kappa , \beta _m, z_0/\delta , L_0/\delta , \Pi , z_c/\delta )$ :

(3.3) \begin{equation} \sqrt {\frac {2}{C_f}}=\frac {1}{\kappa } \left \{\ln {\frac {\delta }{z_{0}}}+\frac {\beta _m z_c/\delta }{L_0/\delta }+\Pi \left [ 2 \left ( \frac {1}{1.2} \right )^2 \left (3-\frac {2}{1.2}\right ) \right ]-\frac {1}{3}\left (\frac {1}{1.2}\right )^2\right \}, \end{equation}

where $\Phi (\delta ,z_c,s)=z_c$ . Taking $\beta _m=8$ and $\kappa =0.41$ , figure 2(c) shows that $\Pi$ increases approximately as a linear function of $\delta _0/L_0$ , and thus is delineated by $\Pi =B_1+B_2\delta /L_0$ , with $B_1\approx 0.485$ and $B_2\approx 0.51$ . In neutral conditions, $\Pi$ approaches $0.485$ ( $0.42$ for $\kappa =0.384$ ). Substituting the above relation and $z_c/\delta =0.185+0.027\delta /L_0$ (figure 2 b) into (3.2) leads to a simplified relationship of $C_f$ with $z_{0}/\delta$ and $L_0/\delta$ for the cases studied:

(3.4) \begin{equation} \sqrt {\frac {2}{C_f}}=\frac {1}{0.41}\bigg[-\ln \left (\frac {z_{0}}{\delta }\right )+0.22\frac {\delta ^2}{L_0^2}+2.42\frac {\delta }{L_0} +0.67\bigg]. \end{equation}

Therefore, a complete velocity profile of a stably stratified boundary layer is determined by (3.2) and (3.4), given $(z_{0}/\delta ,L_0/\delta )$ or $(C_f,L_0/\delta )$ .

Figure 2. (a) Wall-normal profiles of the mean streamwise velocity under varying stable stratification. Solid (dash-dotted) curves are fits of (3.2). The inset shows $\sqrt {C_f/2}$ as a function of $z_0/\delta$ and $\delta /L_0$ . The vertical bar on each symbol represents the difference between the measured value and the prediction from (3.4) (coloured surface). (b) The upper edge of the region under the effect of MOST $z_c/\delta$ . (c) The wake strength $\Pi$ as a function of $\delta /L_0$ . Symbols as in figure 1.

3.3. A modified diffusion analogy

The abrupt roughness change at the surface of a fully developed flow brings at least three distinct effects: (i) streamline displacement, (ii) acceleration/deceleration of flow influencing the mean velocity field (Townsend 1976), and (iii) a local breakdown of the turbulence dynamics leading to an imbalance of turbulence production and dissipation. All three processes can be modulated by the strength of the roughness discontinuity, $M$ . Antonia & Luxton (1971) noted that the roughness change can induce a large vertical gradient of mean streamwise velocity and turbulent intensity, leading to enhanced turbulence production and turbulent diffusion that are predominantly balanced by the advection of turbulent energy. Savelyev & Taylor (2005) incorporated the mean wall-normal velocity into the diffusion analogy by accounting for the streamline displacement, which was formulated as

(3.5) \begin{equation} U\,\frac {{\rm d}\delta _i}{{\rm d}x}=C_1\sigma _w+C_2 W. \end{equation}

We now briefly review the process to derive the IBL formulae in Savelyev & Taylor (2005) in the circumstance of the roughness length, friction velocity and Monin–Obukhov length changing from $(z_{01},u_{*1}, L_{01})$ to $(z_{02},u_{*2}, L_{02})$ . The mean vertical velocity $W$ is approximated as $-\Delta U \delta _i/x$ using the continuity constraint, where $\Delta U$ represents the difference between the local mean streamwise velocity at $z=\delta _i$ and the corresponding value before the roughness change. The expression for $\Delta U$ at $\delta _i$ reads $({u_{*2}}/{\kappa }) (\ln {({\delta _i}/{z_{02}})}+\beta _m ({\delta _i-z_{02}})/{L_{02}})-({u_{*1}}/{\kappa })(\ln {({\delta _i}/{z_{01}})}+\beta _m ({\delta _i-z_{01}})/{L_{01}})$ for neutrally and stably stratified boundary layers. Replacing $u_{*2}$ with $u_{*1}$ simplifies the expression for $W_{{ s}}$ ( $W$ induced by the streamwise displacement) to

(3.6) \begin{equation} W_{{ s}}=\frac {\delta _i}{x}\frac {u_{*1}}{\kappa }\left [M+\beta _m \frac {\delta _i-{z_{01}}}{L_{01}}-\beta _m \frac {\delta _i-{z_{02}}}{L_{02}}\right ]. \end{equation}

In the case of a neutral boundary layer, $L_{01}$ and $L_{02}$ approach infinity. Assuming $\sigma _w/{u_{*1}}=C_0$ , which is reasonable in a constant shear stress layer, the right-hand side of (3.5) can be adjusted to $C_1\sigma _{w1}(1+({C_2}/{C_1}) ({W_s}/{C_0u_{*1})})$ , yielding (36) in Savelyev & Taylor (2005). This formula will hereafter be referred to as the ST model, where $C_1=1,\ C_2=\kappa C_0 \approx 0.51$ ( $C_0\approx 1.25$ ).

To study the impact of the decay in wall-normal stress, we adopt (3.1) and substitute (3.6) into (3.5), giving

(3.7) \begin{equation} \frac {{\rm d}\delta _i}{{\rm d}x}=\frac {a_0 {U_{{n}}}^{-1}-a_1 {U_{{n}}}}{\sigma _L} +C_2\frac {\delta _i}{x}\left ( M+\beta _m \frac {\delta _i-{z_{01}}}{L_{01}}-\beta _m \frac {\delta _i-{z_{02}}}{L_{02}}\right )\frac {\sqrt {C_f}}{\sqrt {2}\,\kappa {U_{{n}}}}. \end{equation}

This is solved by the fourth-order Runge–Kutta algorithm (ode45 in MATLAB), with the integration initiated at $x=z_{01}$ . The initial height of the IBL satisfies $\delta _i(\ln ({{\delta _i}/{\sqrt {z_{01} z_{02}}})}-1)=0.5z_{01}$ (Savelyev & Taylor 2005). Here, we make the assumption that $z_{01}/\delta$ , $\Pi _1$ and $L_{01}/\delta$ are invariant with streamwise location, in line with a fully developed flow. Figure 3(a) benchmarks (3.7) to the dataset, showing a significant improvement in accuracy compared to the ST model.

Figure 3. The IBLs for (a) neutrally stratified boundary layers and (b) stably stratified boundary layers. (a) Cases N1 and N2 with input parameters $(C_f, z_{01}/\delta , M)$ estimated for $\kappa =0.41$ . Symbols: circle, case N1; diamond, case N2. The bluish (yellowish) region with solid (dash-dotted) boundary curves represents the prediction from the ST model. The upper boundary curves in the coloured regions correspond to case N2, and the lower ones to case N1. The red dashed curve represents the prediction for case N1 from the finite-thickness boundary layer model (Li et al. 2022). (b) Cases S1 and S2. Dotted curves represent the prediction of the ST model; dashed curves indicate (3.7); solid curves indicate (3.10). The inset shows data on the upper branch representing $U_{\infty }/U_{\infty ,0}$ , and on the lower branch representing $(W_{\delta }-W_{0})/U_{\infty ,0}$ ; the dashed (solid) lines are linear fits. Symbols as in figure 1.

After validating the model for neutrally stratified boundary layers, we next employ the model (3.7) to study the impact of thermal stability on the IBL development. For cases S1 and S2, Ding et al. (2023) found that the slightly negative mean wall-normal velocity in the outer region (being two orders of magnitude smaller than the free streamwise velocity) could greatly suppress the development of the IBL. The negative wall-normal flow likely originates from the presence of the favourable pressure gradient (Jones et al. 2001). We begin from (3.5) but then consider the wall-normal velocity, noting that this could also be induced by sources other than streamline displacement. To derive $W$ , we rewrite the continuity equation in terms of dimensionless variables $\tilde {x}=x/\delta$ and $\tilde {z}=z/\delta$ using the chain derivative rules $ {{\rm d}U}/{{\rm d}x}=({{\rm d}U}/{{\rm d}\tilde {x}}) ({{\rm d}\tilde {x}}/{{\rm d}x})= ({{\rm d}U}/{{\rm d}\tilde {x}}) ( ({1}/{\delta })-({x}/{\delta ^2})({{\rm d}\delta }/{{\rm d}x}) )$ and ${{\rm d}W}/{{\rm d}z}=({{\rm d}W}/{{\rm d}\tilde {z}})({{\rm d}\tilde {z}}/{{\rm d}z})= ({1}/{\delta })({{\rm d}W}/{{\rm d}\tilde {z}}).$ The integral of ${{\rm d}W}/{{\rm d}\tilde {z}}=- ({{\rm d}U}/{{\rm d}\tilde {x}}) (1-\tilde {x} ({{\rm d}\delta }/{{\rm d}x} ))$ from the upper edge of the IBL to the top of the boundary layer, assuming that ${U_{{n}}}(z/\delta )$ therein remains invariant with streamwise locations, gives

(3.8) \begin{equation} W(x,\delta _i)=W(x,\delta )+\int _{\delta _i/\delta }^{1}\frac {{\rm d}U_{\infty }}{{\rm d}\tilde {x}}\left (1-\frac {x}{\delta }\frac {{\rm d}\delta }{{\rm d}x}\right ){U_{{n}}}(l)\, {\rm d}l, \end{equation}

where $W(x,\delta )$ denotes the mean wall-normal velocity at the top of the boundary layer, $ {{\rm d}U_{\infty }}/{{\rm d}\tilde {x}}$ is the effect of flow acceleration, and the second term in parentheses stands for the effect of the boundary layer thickening process. The inset of figure 3(b) shows that $U_\infty$ and $W$ at the upper edge of the boundary layer both vary with streamwise locations in an approximately linear way. Thus we adopt ${W(x,\delta )}/{U_{\infty ,0}}=w_0+w_1 ({x}/{\delta })$ , and define a constant acceleration coefficient $K= {{\rm d}(U_{\infty }/U_{\infty ,0})}/{{\rm d}\tilde {x}}$ for the cases studied. For both stable cases, $(w_1, K)$ are $(-0.0010,0.0055)$ , and $w_0$ is $-0.0073$ for case S1, and $-0.0133$ for case S2. The flow acceleration observed herein is likely due to the combined effect of the increase in surface roughness and the boundary layer growth on all surfaces due to the fact that the facility does not have an adjustable roof to balance these effects. We note that vertical velocity and flow acceleration have typically been overlooked in previous studies, but a generalisation of the presented model to non-zero pressure gradient is foreseeable. Since the boundary thickening effect is an order of magnitude smaller than the acceleration effect, it can be neglected, simplifying (3.8) to

(3.9) \begin{equation} W_{{ m}}(x,\delta _i)=U_{\infty ,0}\bigg(w_0+w_1 \frac {x}{\delta }+K\int _{\delta _i/\delta }^{1}{U_{{n}}}(l)\, {\rm d}l\bigg). \end{equation}

In the above expression, $W_m$ is estimated from the mean velocity measured at the edge of the boundary layer, and thus contains the effects of acceleration and streamline displacement due to roughness changes. Within the roughness sublayer, (3.9) can break down due to three-dimensional roughness effects, while $W_{{ s}}$ is still effective for predicting the initial growth of the IBL (see ST model performance). Therefore, we replace $W_{{ s}}$ with (3.9) when the top edge of the IBL extends beyond $z_c+3s$ ( $\approx5.5$ roughness element heights, in line with Schultz and Flack 2005). For these reasons, we suggest the following piecewise function:

(3.10) \begin{equation} \displaystyle\frac {{\rm d}\delta _i}{{\rm d}x}= \displaystyle\frac {a_0 U_n^{-1}-a_1 {U_n}}{\sigma _L}+ \begin{cases} \displaystyle\frac {C_2}{{U_{{n}}}(\delta _i)}\displaystyle\frac {\sqrt {C_f}}{\sqrt {2}\,\kappa }\displaystyle\frac {\delta _i}{x}\left ( M+\beta _m \displaystyle\frac {\delta _i-{z_{01}}}{L_{01}}-\beta _m \displaystyle\frac {\delta _i-{z_{02}}}{L_{02}}\right ), &\\[15pt] \qquad \delta _i \leqslant z_c+3s,\\[15pt] \displaystyle\frac {C_2}{{U_{{n}}}(\delta _i)}\left[w_0+w_1 \displaystyle\frac {x}{\delta }+K\int _{\delta _i/\delta }^{1}{U_{{n}}}(l)\, {\rm d}l\right], &\\[15pt] \qquad \delta _i \gt z_c+3s. \end{cases} \end{equation}

Figure 3(b) shows that (3.7) improves the accuracy of the prediction from the ST model, but overprediction still occurs; (3.10) instead well captures the shallow IBL in the outer region. Since the model has no free parameters, it could potentially predict neutrally or stably IBLs in the presence of flow acceleration induced by roughness change or other sources, given the mean velocity at the top of the boundary layer.

4. Conclusions

We have identified two features of stably stratified boundary layers over smooth and transitionally rough surfaces through the analysis of experimental data from a number of wind tunnel studies. First, the wall-normal profiles of outer-scaled mean streamwise velocity in the wake region remain nearly invariant regardless of the stratification, leading to an approximately linear increase in wake strength with increasing $\delta /L_0$ . Second, appropriately scaled profiles of wall-normal fluctuation ( $\sigma _L \sigma _w/U$ ) as a function of $U/U_\infty$ in the cases studied fall onto a unique curve, where the height-independent parameter $\sigma _L-1$ decays exponentially with $\delta /L_0$ . By incorporating these two findings, and the effect of streamline displacement from Savelyev & Taylor (2005), we develop a new diffusion model applicable to the complete growth of the IBLs, irrespective of its depth in relation to the local boundary layer. We then demonstrate the predictive application of this model with experimental data. The new model is found to account adequately for the effect of flow acceleration on the IBL development in the outer region.