Momentum-Flux Determination for Boundary Layers with Sufficient Fetch Based on Integral Equations

Abstract

Integral equations for the vertical velocity component and momentum flux within boundary layers with sufficient fetch under neutral stratifications are derived by extending von Kármán’s integral equation. To simplify the equations, a self-similar profile for the streamwise velocity component and a zero-pressure gradient are assumed. The integral equations enable the following calculations. The vertical velocity component and momentum flux are calculated by the integral of the streamwise change in the streamwise velocity component. The vertical velocity component at the boundary-layer top is determined by the drag coefficient and shape factor. The advective flux at the boundary-layer top is of the same order as the turbulent momentum flux near the surface. Accordingly, the derived integral equations are applied to various types of boundary layers with sufficient fetch. For laminar boundary layers, profiles determined by the equations perfectly agree with the profiles determined by the analytical Blasius solution. For turbulent boundary layers, the comparison of the momentum flux between the proposed equations and experimental data verifies that the equations predict the turbulent momentum flux very well. Finally, we employ a power-law approximation for determining both the velocities and the momentum flux. As the streamwise velocity component can be approximated by the power law for both smooth and rough surfaces, the resulting turbulent flux agrees well with that obtained in experiments and numerical simulations.

Appendices

Appendix 1: Integral Equation of the Vertical Velocity Component

According to Gauss’s theorem, the volume integral of Eq. 1 with respect to $V$ in Fig. 1 is expressed by following surface integrals with the definition of influx as positive:

$$ \mathop \int \limits_{V} \frac{{\partial u_{i} }}{{\partial x_{i} }}dV = \underbrace {{\mathop \int \limits_{{S_{xr} - S_{xl} }} udS_{x} }}_{{\text{I}}} - \underbrace {{\left( { - \mathop \int \limits_{{S_{xb} }} udS_{x} + \mathop \int \limits_{{S_{zb} }} wdS_{z} } \right)}}_{{{\text{II}}}} + \underbrace {{\left( { - \mathop \int \limits_{{S_{xt} }} udS_{x} + \mathop \int \limits_{{S_{zt} }} wdS_{z} } \right)}}_{{{\text{III}}}} = 0 $$

(37)

The integrals on $S_{b}$ and $S_{t}$ are considered negative and positive, respectively. On each surface integral, the integral with respect to ${S}_{x}$ is defined with a negative sign because of the direction of the surface [This is confirmed by the following: when the surface curve is denoted as $z={z}_{c}\left(x\right)$, the unit normal vector pointing away from the curve is ${\varvec{n}}={L}^{-1}\left(-d{z}_{c}/dx, 1\right)$, where $L=\sqrt{1+{\left(d{z}_{c}/dx\right)}^{2}}$.] By following the derivation of von Kármán’s integral equation, term I can be expressed as

$${\int }_{{S}_{xr}-{S}_{xl}}ud{S}_{x}={\int }_{{z}_{b}\left(x+\Delta x\right)}^{{z}_{c}\left(x+\Delta x\right)}udz-{\int }_{{z}_{b}\left(x\right)}^{{z}_{c}\left(x\right)}udz=\frac{d}{dx}\left({\int }_{{z}_{b}}^{{z}_{c}}udz\right)\Delta x.$$

(38)

Term II on the r.h.s. of Eq. 37 becomes zero because of the surface integrals at the bottom boundary with $u({z}_{b})=0$ and $w({z}_{b})=0$. The second term in term II does not appear when the bottom surface is smooth, or $\mathrm{when} {z}_{b}(x)=0$. Term III on the r.h.s. of Eq. 37 represents the surface integrals at the top boundary expressed by the arbitrary curve ${z}_{c}$. At the top boundary, the constraint condition $z={z}_{c}$ allows for the following conversion of integrals

$$-{\int }_{{S}_{xt}}ud{S}_{x}+{\int }_{{S}_{zt}}wd{S}_{z}=-{\int }_{x}^{x+\Delta x}u\left(x,{z}_{c}\right)\frac{d{z}_{c}}{dx}dx+{\int }_{x}^{x+\Delta x}w\left(x,{z}_{c}\right)dx.$$

(39)

Since the integral between small differentials yields the original integrand when $\Delta x\to 0$, Eq. 39 becomes

$$-{\int }_{{S}_{xt}}ud{S}_{x}+{\int }_{{S}_{zt}}wd{S}_{z}=-u\left(x,{z}_{c}\right)\frac{d{z}_{c}}{dx}\Delta x+w\left(x,{z}_{c}\right)\Delta x$$

(40)

Therefore, Eq. 37 leads to Eq. 4.

Appendix 2: Integral Equation of the Streamwise Momentum Equation

According to Gauss’s theorem, the volume integral of Eq. 2 with respect to control volume $V$ in Fig. 1 is expressed as

$$ \mathop \int \limits_{V} \left( { - \frac{{\partial uu_{j} }}{{\partial x_{j} }} + \nu \frac{{\partial^{2} u}}{{\partial x_{j} \partial x_{j} }} - \frac{{\partial \widetilde{{p_{0} }}}}{\partial x}} \right)dV = \underbrace {{\mathop \int \limits_{{S_{xr} - S_{xl} }} \left( { - uu + \nu \frac{\partial u}{{\partial x}} - \tilde{p}} \right)dS_{x} }}_{{\text{I}}} - \underbrace {{\left( { - \mathop \int \limits_{{S_{xb} }} \left( { - uu + \nu \frac{\partial u}{{\partial x}} - \tilde{p}} \right)dS_{x} + \mathop \int \limits_{{S_{zb} }} \left( { - uw + \nu \frac{\partial u}{{\partial z}}} \right)dS_{z} } \right)}}_{{{\text{II}}}} + \underbrace {{\left( { - \mathop \int \limits_{{S_{xt} }} \left( { - uu + \nu \frac{\partial u}{{\partial x}} - \tilde{p}} \right)dS_{x} + \mathop \int \limits_{{S_{zt} }} \left( { - uw + \nu \frac{\partial u}{{\partial z}}} \right)dS_{z} } \right)}}_{{{\text{III}}}} = 0 .$$

(41)

Here, the storage term $\partial u/\partial t$ is assumed to be zero because the term does not contribute to the temporally averaged momentum budget. Term I on the r.h.s. can be converted to a derivative with respect to $x$ in the same manner as in Eq. 38.

Term II on the r.h.s. of Eq. 41 represents the surface integrals at the bottom boundary. Owing to the boundary condition of $u({z}_{b})=0$ and $w\left({z}_{b}\right)=0$, the first term of each integrand, namely the advective flux term, becomes zero, whereas the other terms indicate a drag force acting on the surface as

$$-{\int }_{{S}_{xb}}\left(-uu+\nu \frac{\partial u}{\partial x}-\tilde{p }\right)d{S}_{x}=-{\int }_{x}^{x+\Delta x}\left(\nu \frac{\partial u\left(x,{z}_{b}\right)}{\partial x}-\tilde{p }\left(x,{z}_{b}\right)\right)\frac{d{z}_{b}}{dx}dx =\frac{{\tau }_{norm}}{\rho }\Delta x,$$

(42)

$${\int }_{{S}_{zb}}\left(-uw+\nu \frac{\partial u}{\partial z}\right)d{S}_{z}={\int }_{x}^{x+\Delta x}{\left[\nu \frac{\partial u}{\partial z}\right]}_{z={z}_{b}}dx =\frac{{\tau }_{fric}}{\rho }\Delta x,$$

(43)

where ${\tau }_{norm}$ represents streamwise compression due to viscosity and pressure form drag per unit area, and ${\tau }_{fric}$ indicates surface friction drag. The force acting on the solid surface in the streamwise direction is defined as the positive value. For a smooth surface, or ${z}_{b}=0$, only ${\tau }_{fric}$ appears, whereas both ${\tau }_{norm}$ and ${\tau }_{fric}$ need to be considered for rough surfaces. For a smooth or rough surface, the total friction drag is denoted by ${u}_{*}$ as $\tau ={\tau }_{norm}+{\tau }_{fric}=\rho {u}_{*}^{2}$.

Term III on the r.h.s. of Eq. 41 represents the surface integrals at the top boundary. Following the derivation of Eqs. 39 and 40, the term is expressed as

$$-{\int }_{{S}_{xt}}\left(-uu+\nu \frac{\partial u}{\partial x}-\tilde{p }\right)d{S}_{x}+{\int }_{{S}_{zt}}\left(-uw+\nu \frac{\partial u}{\partial z}\right)d{S}_{z}=\left(-{\left[-uu+\nu \frac{\partial u}{\partial x}-\tilde{p }\right]}_{{z}_{c}}\frac{d{z}_{c}\left(x\right)}{dx}+{\left[-uw+\nu \frac{\partial u}{\partial z}\right]}_{{z}_{c}}\right)\Delta x.$$

(44)

Finally, Eq. 6 was obtained using Eqs. 41–45.

Appendix 3: Consistency with von Kármán’s Integral Equation

von Kármán’s integral equation (Schlichting and Gersten 2000) is written as

$${u}_{*}^{2}=\frac{d}{dx}\left(\theta {u}_{0}^{2}\right)+{u}_{0}\frac{d{u}_{0}}{dx}{\delta }^{*}.$$

(45)

Here, $\theta$ and ${\delta }^{*}$ represents the momentum and displacement thicknesses defined as

$$\theta ={\int }_{0}^{\delta }\frac{u}{{u}_{0}}\left(1-\frac{u}{{u}_{0}}\right)dz,$$

(46)

$${\delta }^{*}={\int }_{0}^{\delta }\left(1-\frac{u}{{u}_{0}}\right)dz.$$

(47)

The integral equation of Eq. 8 is consistent when the integral range is extended to ${z}_{c}=\delta$. By substituting ${z}_{c}=\delta$ into Eqs. 5 and 8, we obtain

$${w}_{0}=-{\int }_{0}^{\delta }\frac{\partial u}{\partial x}dz$$

(48)

$$-{u}_{0}{w}_{0}={u}_{*}^{2}+\delta \frac{d\stackrel{\sim }{{p}_{0}}}{dx}+{\int }_{0}^{\delta }\frac{\partial {u}^{2}}{\partial x}dz.$$

(49)

Here, ${w}_{0}=w\left(x,\delta \right)$, and $d{u}_{0}/dz=0$. In addition, $\nu \partial u/\partial x$ is neglected when the molecular diffusion is sufficiently smaller than $\partial {u}^{2}/\partial x$. By substituting ${w}_{0}$ of Eq. 49 into Eq. 50, we obtain

$${u}_{*}^{2}={\int }_{0}^{\delta }\left(\frac{\partial {u}_{0}u}{\partial x}-u\frac{d{u}_{0}}{dx}\right)dz-{\int }_{0}^{\delta }\frac{\partial {u}^{2}}{\partial x}dz-\delta \frac{d\stackrel{\sim }{{p}_{0}}}{dx}.$$

(50)

By employing the Leibnitz theorem and Eq. 3, we get

$${u}_{*}^{2}={\int }_{0}^{\delta }\frac{\partial }{\partial x}\left({u}_{0}^{2}\left(\frac{u}{{u}_{0}}\right)\left(1-\frac{u}{{u}_{0}}\right)\right)dz-\frac{d\stackrel{\sim }{{p}_{0}}}{dx}{\int }_{0}^{\delta }\left(1-\frac{u}{{u}_{0}}\right)dz=\frac{d}{dx}\left({u}_{0}^{2}\theta \right)+{u}_{0}\frac{d{u}_{0}}{dx}{\delta }^{*}.$$

(51)

Appendix 4: Consistency with Equation for Laminar Boundary Layers

Blasius derived a universal function $f(\eta )$ with a single scaling variable $\eta =z\sqrt{{u}_{0}/\nu x}$ for a developing two-dimensional laminar boundary layer (Schlichting and Gersten 2000). Velocity components $u$ and $w$ with two variables $(x, z)$ are reduced to one-variable functions of $\eta$ by introducing the streamfunction expressed as $\phi (x,\eta )=\sqrt{{u}_{0}\nu x}f(\eta )$. According to the definition of the streamfunction, streamwise and vertical velocity components are expressed as

$$u=\frac{\partial \phi }{\partial z}={u}_{0}\frac{df}{d\eta },$$

(52)

$$w=-\frac{\partial \phi }{\partial x}=-\frac{1}{2}\sqrt{\frac{{u}_{0}\nu }{x}}\left(f\left(\eta \right)-\eta \frac{df}{d\eta }\right).$$

(53)

In the Blasius solution, a universal function $f(\eta )$ exists; therefore, the boundary-layer depth is universally expressed as $\delta ={\eta }_{\delta }/\sqrt{{u}_{0}/\nu x}$, where ${\eta }_{\delta }$ is a constant value determined by a universal function (${\eta }_{\delta }\approx 5$, (Schlichting and Gersten 2000)). Thus, $\eta$ is expressed by a boundary-layer scaling variable $\zeta =z/\delta$ as $\eta ={\eta }_{\delta }\zeta$. Using Eq. 53, Eq. 54 normalized by ${u}_{0}$ as a function of ${\zeta }_{c}={z}_{c}/\delta$ is written as

$${g}_{w}\left({\zeta }_{c}\right)=\frac{d\delta }{dx}{\int }_{{\zeta }_{b}}^{{\zeta }_{c}}\left({g}_{u}\left({\zeta }_{c}\right)-{g}_{u}\left(\zeta \right)\right)d\zeta .$$

(54)

Here, $0.5 \delta /x=d\delta /dx$ for the laminar boundary layer. Hence, Eq. 54 is consistent with Eq. 22.

Appendix 5: Experimental Details

To obtain the streamwise velocity component and turbulent momentum-flux profiles over a simplified cubical block array, we conducted wind-tunnel experiments to measure the instantaneous two-directional components of velocities.

Experimental facilities and designs were those suggested by Hagishima et al. (2009) who reported the geometric dependency of the drag coefficient for various types of simplified arrays using a circuit-type wind tunnel in the Faculty of Engineering Science, Kyushu University, Japan. The details of the wind-tunnel device can be found in Hagishima et al. (2009).

The schematics of the measurement details and picture of a sensor installation are shown in Fig. 8. We adopted a cubical block array arranged in a staggered layout with a packing density ${\lambda }_{p}$ of 17% (named ST1 in their paper). The block height $h$ was 25 mm. The fetch was approximately $137h$, which made $\delta$ around $6h$. The reference wind speed ${u}_{ref}$, which was approximately 4 m s⁻¹, was measured at $z=20h$. The measurement locations in each unit and spanwise location are shown in Fig. 8b. The measurements were conducted at four locations from A to D in a unit and five units of the first to fifth unit in the spanwise direction to obtain the representative spatially-averaged profiles. In the vertical direction, the lowest positions were $1.6h$ and $0.8h$ at A and B, and at C and D, respectively. The vertical measurement intervals were $0.1h$ (2.5 mm) below $z=8h$. Above $8h$, which is out of the boundary layer, the velocities were measured at three heights of $10h,$ $15h$, and $20h$. Data used in analyses were spatially-averaged data from all locations.

Laser doppler anemometry (LDA) was adopted for measurement using FiberFlow 2D and a BSA F/P30 processor (Dantec Dynamics Ltd.), which enable us to detect two-velocity components using a dual-laser beam from a probe with focal of 160 mm. Figure 8c shows the probe and the blocks. The probe was attached to a three-dimensional traverse system in the wind tunnel and the locations are automatically controlled during the series of experiments. Because the sampling frequency varies with the number of oil particles that passed through the focal position of the laser, we cannot fix the frequency. However, we confirmed that the sampling frequency was approximately 1000 Hz for the measurement period of 30 s after the measurements. The oil particles were generated by using PIVPart 14 (Seika Digital Image), which employed the Laskin nozzle particle generator. The particle diameters were approximately 1 μm. The particles were inserted into the wind tunnel at an upstream position of the cubical block array from a small hole on the bottom floor of the tunnel. The particle concentration was adjusted based on the sampling frequency of LDA to obtain a sufficient sampling frequency of 1000 Hz. The friction velocity was determined by the peak value of the Reynolds stress averaged in all the measurement positions in horizontal plain.

Appendix 6: Application to Fully-Developed Flow

The simplest case to apply integral equations is the condition of horizontally homogeneous flow. The condition is precisely achieved in a fully developed boundary layer such as pipe flow driven by a pressure gradient and Couette flow driven by a top-moving wall. These conditions are commonly adopted, with periodic conditions, in numerical simulations that reproduce infinitely repeated surface conditions (Watanabe 2004; Coceal et al. 2006). It is analytically known for a fully developed boundary layer that i) the vertical velocity component is precisely zero in the entire regions, ii) the advective momentum flux is zero, and iii) the turbulent momentum flux is analytically expressed by a constant value for Couette flow and a linear-decreasing function for pipe flow; therefore, these facts must be consistent in the derived integral equations.

In a fully-developed flow, a streamwise velocity component and the boundary-layer depth are also kept constant. Hence, we can easily obtain the following relations

$$\overline{{g}_{w}}\left({\zeta }_{c}\right)=\frac{d\delta }{dx}{\int }_{{\zeta }_{b}}^{{\zeta }_{c}}\zeta \frac{d\overline{{g}_{u}}}{d\zeta }d\zeta =0,$$

(55)

$$ \overline{{F_{t} }} \left( {\zeta _{c} } \right) = C_{d} + \frac{{\zeta _{c} \delta }}{{u_{0}^{2} }}\frac{{d\tilde{p}_{0} }}{{dx}} + \frac{{d\delta }}{{dx}}\smallint _{{\zeta _{b} }}^{{\zeta _{c} }} \zeta \frac{{d\overline{{g_{u} }} ^{2} }}{{d\zeta }}d\zeta = C_{d} + \frac{{\zeta _{c} \delta }}{{u_{0}^{2} }}\frac{{d\tilde{p}_{0} }}{{dx}} .$$

(56)

In both equations, terms with $d\delta /dx$ vanished because $\delta$ is constant. Because of this, $\overline{{g}_{w}}=0$, thereby yielding ${F}_{a}=0$ and $\overline{{F}_{t}}={F}_{d}$. Although we can replace the pressure term by ${u}_{0}$ in Eq. 3, the pressure term must be expressed by $d\stackrel{\sim }{{p}_{0}}/dx$ in Eq. 56 because ${u}_{0}$ cannot be defined in a fully-developed flow.

When Couette flow is driven under the zero-pressure gradient, we simply obtain $\overline{{F}_{t}}={C}_{d}$. That is, a constant-flux layer can be achieved. For pipe flow, we obtained $\overline{{F}_{t}}={C}_{d}(1-\zeta )$ because $d\stackrel{\sim }{{p}_{0}}/dx=-{u}_{*}^{2}/\delta$. Both these profiles are consistent with the aforementioned facts for two types of fully-developed flow.

In various previous studies dealing with the turbulent boundary layer over urban-like arrays or plant canopies, periodic boundary conditions were imposed in the lateral and streamwise directions to reduce computational costs (for vegetation canopy: Watanabe 2004; Finnigan et al. 2009; for urban canopy: Kanda 2006; Xie and Castro 2006; Coceal et al. 2006; Coceal et al. 2007; Kono et al. 2010; Abd. Razak et al. 2013; Boppana et al. 2014; Ikegaya et al. 2016, 2017). This requires artificial momentum sources such as pressure gradient or moving top boundary to drive the flow in the numerical domain. However, such boundary conditions force the vertical velocity component to be zero, and the momentum provision uniquely determine the turbulent momentum-flux profile. Both these resultant flow characteristics differ from those of boundary layers discussed in Sect. 3. This means that numerical simulations with periodic boundary conditions cannot provide results of momentum flux comparable with those of the wind-tunnel experiments or field measurements over urban or vegetation canopies. Although it is unclear how significantly these differences in driving forces affect flow and turbulent structures in the boundary layers, the momentum flux similarity must not be fulfilled precisely between boundary layers with sufficient fetch and such numerical simulations.