Experimental Study on Flow Structure Characteristics of Gap Flow Boundary Layer Based on PIV

Abstract

The boundary layer is the main source of frictional resistance in gap flow, and the study of the flow structure characteristics of the gap flow boundary layer is of great significance for the study of gap flow theory. In this study, the PIV technique was utilized to experimentally investigate the gap flow boundary layers with Reynolds numbers of 16,587–56,870 and gap ratios of 0.6–0.8. The characteristics of the wall friction velocity, the boundary layer thickness, and the wall function of the gap flow boundary layer were analyzed, and the influences of the mean velocity of the gap flow and the gap ratio on the flow structure characteristics of the boundary layer were explored. The results show that using PIV to measure the velocity profile in the viscous sub-layer to solve for the wall friction velocity had good precision. The boundary layer thickness was inversely proportional to the mean velocity of the gap flow and the gap ratio. The wall functions of the boundary layer were as follows: in the viscous sub-layer ($y^{+}<$ 5.5), $u^{+}=y^{+}$; in the transition layer (5.5 $<y^{+}<$ 26), $u^{+}=\frac{1}{0.071}tanh\left(0.071y^{+}\right)$; and in the logarithmic layer ($y^{+}>$ 26), $u^{+}=2.78ln\left(y^{+}\right)+3.8$. The thickness of the logarithmic layer was proportional to the mean velocity of the gap flow and inversely proportional to the gap ratio. The inner region of the boundary layer extended to $y<$ 0.18$\delta$ or $y<$ 0.13($h$/2).

1. Introduction

Gap flow refers to the flow phenomenon that occurs when a liquid passes through a gap between two parallel or near-parallel surfaces. There are gaps in valves, pumps, pipelines, and other components in a hydraulic system, and the existence of gap flow has a certain impact on the performance, safety, and reliability of the hydraulic system, for example, it will cause leakage, reduce the sealing effect, and affect the motion characteristics and energy efficiency of components. Therefore, scholars have conducted many studies on the characteristics of gap flow. Lu et al. [1] investigated the characteristics of the gap flow between the valve core and the valve sleeve of a two-dimensional servo valve and studied the effect of the gap flow on the frictional resistance using theoretical and simulation methods in order to achieve a larger power-to-weight ratio for the servo valve. To predict the hydrodynamic performance of a pump jet propulsor more accurately, Hu et al. [2] developed a gap flow model by considering the influence of the rotor tip gap flow based on potential flow theory and presented the optimal height range of the gap model. Peng et al. [3] studied the influence of a gap between an impeller and pump cover of a centrifugal pump on the hydraulic performance using a combination of experiments and numerical simulations and drew the conclusion that with an increase in the gap height, the impact loss at the tongue was significantly improved and the effect of water flowing out from the gap interfering with the water at the impeller inlet gradually improved, which has a significant guiding effect for the design of the gap between the impeller and cover in the future. Jia et al. [4] analyzed the effects of the Reynolds number on the velocity distribution and pressure distribution of the gap flow in a hydraulic pipeline through simulations and experiments to provide a theoretical basis for reducing the energy loss of a hydraulic pipeline transport system.

In turbulence theory, if the turbulent structure is directly affected by the solid boundary wall, this turbulence is called wall turbulence [5]. Gap flow is a form of wall turbulence, and thus, the gap flow can be divided into two regions: One is the region near the wall, where the flow is not affected by the flow away from the wall and the wall conditions are expressed by the wall shear stress. In this region, viscous forces gradually change from a dominant role to the same order of magnitude as the inertial forces; this region is called the boundary layer. In another region outside this region, the flow is not affected by the wall and the fluid motion is mainly governed by inertial forces; this region is called the potential flow.

The concept of a boundary layer was first proposed by Prandtl for infinite bypass turbulence, where he assumed that the pressure value did not change throughout the boundary layer, that is, the pressure distribution on the borderline of the boundary layer was equal to the pressure distribution within the boundary layer, which was equal to the pressure distribution of the ideal fluid bypassing the flow. Then, a set of differential equations for the boundary layer was obtained by magnitude analysis and simplification of N-S equations:

$$\left\{\begin{array}{c}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\\ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial x}+\nu \frac{{\partial }^{2}u}{\partial y^2}\end{array}\right.$$

(1)

where the boundary conditions and initial conditions are $y=0, u=v=$0; $y=\delta , u=U(x,t)$, where $U(x,t)$ is the velocity distribution of the ideal fluid bypassing the flow; $t=t_0$, the distributions of $u$ and $v$ are known. Prandtl’s system of boundary layer equations, even after simplification, was still a second-order nonlinear system of partial differential equations, and it was still quite difficult to find their theoretical solutions. For this reason, people had to use approximation methods. In 1921, Von Kármán, who was a student of Prandtl, proposed the method of momentum integral relativistic formulas, which was known as similarity theory [6], that is, an approximate velocity distribution in the boundary layer $u\left(x\right)$ was assumed to replace the real velocity distribution $u(x,y)$, and it was made to satisfy the momentum integral relation and the boundary conditions on average and in general without requiring that the motion of every fluid mass satisfy the boundary layer differential equation. This was the famous Kármán’s momentum integral equation:

$$\frac{d}{dx}\left({\int }_0^{\delta }\frac{u}{U}(1-\frac{u}{U})dy\right)+\left({\int }_0^{\delta }\frac{u}{U}(1-\frac{u}{U})dy\right)\frac{2}{U}\frac{dU}{dx}+\left({\int }_0^{\delta }(1-\frac{u}{U})dy\right)\frac{1}{U}\frac{dU}{dx}=\frac{{\tau }_w}{\rho U^2}$$

(2)

where $\rho$ is the density of the fluid; $U$ can be solved using potential flow theory; and the unknown quantities are $u,\delta ,$ and ${\tau }_w$, where $\delta$ is the thickness of the boundary layer, ${\tau }_w$ is the wall shear stress, and $u$ is the velocity in the boundary layer.

In the studies of the boundary layer, the flow structure of the boundary layer is generally divided into an inner region and an outer region. Within these two regions, the wall friction velocity is usually taken as the velocity characteristic quantity, which is defined as

$$u_{*}=\sqrt{\frac{{\tau }_w}{\rho }}$$

(3)

In 1933, Prandtl [7] proposed that the velocity distribution in the inner region depends on the distance from the wall, the wall shear stress, and the characteristics of the fluid, i.e.,

$$u=f_i(y,{\tau }_w,\rho ,\mu ),$$

(4)

where $f_i$ denotes the functional relationship, $y$ is the distance from the wall, and $\mu$ is the dynamic viscosity of the fluid.

Dimensional analysis of Equation (4) leads to

$$u^{+}=f\left(y^{+}\right),$$

(5)

where $u^{+}=\frac{u}{u_{*}}$ and $y^{+}=\frac{y{u}_{*}}{\nu }$. Equation (5) is called the wall function.

The logarithmic law is a commonly used wall function, which is a logarithmic function of the velocity distribution obtained from the analysis of turbulence theory, namely,

$$u^{+}=\frac{1}{k}ln{y}^{+}+B,$$

(6)

where $k$ is von Kármán’s constant and $B$ is a constant. In boundary layer flows, in addition to the application of the logarithmic velocity distribution, a purely empirical power distribution is often utilized, namely,

$$u^{+}=m{\left(y^{+}\right)}^{n },$$

(7)

where $m$ is a constant and $n$ is the exponent.

In 1932, Nikuradse [8] published the results of his experimental study of turbulence in a smooth circular pipe, and over the past half-century, there has been a growing body of research literature on the structural characteristics of the boundary layer. Zhao [9] conducted experiments and numerical simulations of the average velocity distribution, logarithmic law, and von Kármán’s constant for turbulence in a circular pipe, and concluded that the constant $k$ in the logarithmic law was a function of the Reynolds number or Kármán’s number. Zagarola [10] considered the existence of two regions in a pipe flow, with one satisfying an exponential law distribution and another satisfying a logarithmic law distribution, and that the values of both the constants $\mathrm{k}$ and B in the logarithmic law were independent of the Reynolds number. Dong [11] studied the open-flow flat plate boundary layer and found that the development of its thickness could be quoted from the formulas for the infinite bypass flat plate boundary layer, and the velocity distribution could be expressed in logarithmic form, where the constant $\mathrm{k}$ was 0.39, or in exponential form, where the exponent was related to the Reynolds number; in contrast, the boundary layer on the surface of a spillway dam or the bucket of an outlet structure [12,13], the exponential velocity distribution had a larger adaptive range and fitness than that of the logarithmic distribution. After the 1950s, with the rapid development of computational fluid dynamics (CFD) [14,15,16], direct numerical simulation (DNS) has become a valuable tool for studying the boundary layer. Kim [17] and Moser [18] used DNS to study the flow in a fully developed rectangular channel and found that the larger the Reynolds number, the smaller the constant B of the logarithmic law. Wu [19] and Xu [20] also used DNS to calculate the velocity distributions of the circular pipe turbulence and channel turbulence, respectively, and found that the two were more different in the logarithmic region. However, due to computational limitations, DNS can only simulate low-Reynolds-number flows.

Despite the significant effort in the boundary layer flows, there is poor agreement between the reported results. The reason may be due to the experimental uncertainties involved in measuring the turbulence quantities near the wall, where the presence of high shear and small-scale turbulent motions makes the measurement extremely difficult. Johansson [21] reported the effect on turbulence measurements of imperfect spatial resolution due to probe length. Zhang [22] used laser Doppler velocimeter (LDV) technology to measure the average velocity profile of a smooth flat boundary layer and found that the laser control body of the LDV was an ellipsoidal spot, which could not be moved to the flat wall to determine the position of the wall. Zhang [23] conducted an experimental study on the velocity boundary layer of a rectangular channel using the particle image velocimetry (PIV) technique. Yao et al. [24] carried out an experimental study on the inlet plugging of a rectangular narrow gap channel using the PIV technique. Their results showed that the tracer particles could better reflect the flow characteristics in the rectangular channel. Therefore, in this study, for the fully developed gap flow, PIV technology was used to experimentally investigate the flow structure characteristics of the boundary layer to further understand the flow characteristics of the boundary layer, improve gap flow theory, and optimize the design and performance of the hydraulic system.

2. Experimental Program

2.1. Experimental System

The experimental system mainly consisted of several parts, such as the power device, water pipeline, flow-stabilizing device, test section, and receiving device. The arrangement of the experimental system is shown in Figure 1. The power device was a centrifugal pump and the flow rate was controlled using a regulator valve and an electromagnetic flow meter. The water pipeline was a PPR round pipe with a length of 15 m. The flow-stabilizing device consisted of three parts: a gradual expansion section, a flow-stabilizing section, and a gradual contraction section, among which the length of both the gradual expansion section and the gradual contraction section was 140 mm and the length of the stabilizing section was 1 m. The test section was a Plexiglas square pipe with a cross-section size of 50 × 50 mm² and a length of 8.2 m. The receiving device was a stainless steel tank, which was also the water-supplying device for the whole system.

2.2. Measuring Device

The measurement devices of this experiment mainly included a pressure measurement device and a flow velocity measurement device. The pressure measurement device was mainly composed of two parts: data acquisition equipment and dynamic signal continuous monitoring system software V3.0, as shown in Figure 2.

A particle image velocimetry (PIV) system was selected for the flow velocity measurement, as shown in Figure 3. In order to effectively eliminate the interference caused by wall scattering, the test section was wrapped with black paper, except for the laser-incident position. The tracer particles used in the experiment were hollow glass microbeads with a density of 1.1 g/cm³ and a particle size of 3–6 μm.

2.3. Experimental Program Design

2.3.1. Test Conditions

As shown in Figure 4, the gap channel was formed by the flat plate and the square pipe, and in this study, three heights of flat plates were chosen to produce three gap ratios $\beta \left(\beta =\frac{a}{b}\right)$. The specific parameters are shown in

Table 1. Parameters of gap channels.

Gap Ratio $\mathit{\beta }$	Plate Height $\mathit{a}/\mathrm{mm}$	Square Pipe Height $\mathit{b}/\mathrm{mm}$	Square Pipe Width (Gap Width) $\mathit{c}/\mathrm{mm}$	Gap Height $\mathit{h}/\mathrm{mm}$
0.6	30	50	50	20
0.7	35	50	50	15
0.8	40	50	50	10

. The mean velocities of the gap flow $\left(u_m\right)$ were 1 m/s, 1.5 m/s, and 2 m/s. The three mean velocities and three gap ratios resulted in nine Reynolds numbers, as shown in

Table 2. Reynolds numbers of gap flow.

Gap Ratio $\mathit{\beta }$	Mean Velocity of Gap Flow ${\mathit{u}}_{\mathit{m}/m/s}$	Reynolds Number of Gap Flow $\mathit{R}\mathit{e}$
0.6	1	28,435
	1.5	42,652
	2	56,870
0.7	1	22,967
	1.5	34,450
	2	45,933
0.8	1	16,587
	1.5	24,880
	2	33,174

. The length of the flat plate was 2.4 m.

2.3.2. Arrangement of Test Sections and Test Lines

When the flow entered the gap, the cross-section reduced and the hydraulic forces here inevitably changed instantaneously, resulting in the redistribution of the original stable flow field in the gap. Therefore, the flow in the gap could be divided into two sections along the flow direction: the developing section and the fully developed section. In the developing section, due to the influence of the inlet, the flow velocity distribution, the shear stress distribution, and the pressure distribution changed significantly with the flow position. Therefore, the boundary layer flow in this region was very different from those in the fully developed section. The tests of this experiment were mainly for the fully developed gap flow. Taking into account all the mean velocities of the gap flow and the gap ratios in this study, the selection of the test sections for the fully developed gap flow should be more than 1.8 m away from the inlet. Therefore, the test sections were arranged at distances of 1.8 m, 1.9 m, 2.0 m, 2.1 m, and 2.2 m from the inlet; the average of the results of the five test sections was taken as the final result. The arrangement of the test sections is shown in Figure 5a.

From the literature [25], it can be seen that the gap flow field had symmetry in both the x-direction and y-direction. Therefore, in this study, the boundary layer on the upper wall of the flat plate was taken as the research object. In order to form the boundary layer to be studied on the upper wall of the flat plate, it was necessary to ensure that the gap flow had two-dimensional properties, i.e., the sidewalls of the gap channel did not affect the flow in the central region. Therefore, the vertical centerline in the test section (i.e., $\mathrm{x}=0$) was taken as the first test line, and the other four test lines were arranged at 5 mm intervals along the positive x-direction. The arrangement of the test lines is shown in Figure 5b.

The two-dimensionality of the gap flow was verified here. The velocity distributions of different test lines with the mean velocity of 1 m/s and the gap ratios of 0.6, 0.7, and 0.8 are shown in Figure 6, where $\mathrm{y}$ is the normal distance from the measuring point to the upper wall of the flat plate and $h$ is the height of the gap. The results show that the velocity distribution of the gap flow remained constant within 60% of the gap width, while the velocity distributions near the sidewall were low due to the influence of the angular region, and thus, the gap flow investigated in this study had two-dimensional properties.

3. Results

3.1. Calculation of Wall Friction Velocity

For the incompressible two-dimensional fully developed turbulent flow, the balance of forces can be expressed as

$$\frac{\partial p}{\partial z}·\frac{h}{2}={\tau }_w$$

(8)

where $\frac{\partial p}{\partial z}$ is the pressure gradient in the flow direction, $h$ is the height of the gap flow, and ${\tau }_w$ is the wall shear stress.

The wall shear stress can be expressed as

$${\tau }_w={\mu \left(\frac{du}{dy}\right)}_{y=0}$$

(9)

The wall friction velocity can be calculated using

$$u_{*}=\sqrt{\frac{{\tau }_w}{\rho }}$$

(10)

where $u_{*}$ is the wall friction velocity, $\mu$ is the dynamic viscosity coefficient, and $\rho$ is the density.

From the above equations, it can be seen that in a two-dimensional fully developed turbulent flow, the pressure is linearly distributed along the flow direction. Therefore, two methods were used to calculate the friction velocity in this experiment. The first method used Equations (8) and (10) to calculate $u_{*}$ based on the pressure measurement results; the second method used Equations (9) and (10) to calculate $u_{*}$ based on the measured velocity profile in the viscous sub-layer. The calculation results of the two methods are shown in

Table 3. Values of $u_{*}$ calculated using the two methods.

$\mathit{\beta }$ = 0.6
	$u_m$ =1 m/s	$u_m$ = 1.5 m/s	$u_m$ = 2 m/s
Method 1	0.0568	0.0820	0.1056
Method 2	0.0549	0.0781	0.1004
Relative error	3.35%	4.76%	4.92%
$\mathit{\beta }$ = 0.7
	$u_m$ = 1 m/s	$u_m$ = 1.5 m/s	$u_m$ = 2 m/s
Method 1	0.0577	0.0836	0.1084
Method 2	0.0567	0.0806	0.1035
Relative error	1.73%	3.59%	4.52%
$\mathit{\beta }$ = 0.8
	$u_m$ = 1 m/s	$u_m$ = 1.5 m/s	$u_m$ = 2 m/s
Method 1	0.0580	0.0838	0.1088
Method 2	0.0577	0.0817	0.1049
Relative error	0.52%	2.51%	3.58%

.

As can be seen from Table 3, the calculation results of the two methods were very similar, and the maximum relative error did not exceed 5%, which indicates that the experimental program in this study was feasible, and using PIV to measure the velocity profile in the viscous sub-layer to solve for the wall friction velocity had good precision. The wall friction velocity increased with the increase in the mean velocity of the gap flow and increased with the increase in the gap ratio. From the DNS data obtained by previous authors [26], it was found that the velocity distribution with the normal distance was not strictly linear in the viscous sub-layer and the slope of its tangent line $\frac{du}{dy}$ decreased as $y^{+}$ increased. Due to the small number of particles close to the wall, the location of the velocity-measuring point closest to the wall fell in the part of the viscous sub-layer where the slope of the tangent line was small, which led to the values of ${\mathrm{u}}_{*}$ calculated by the second method being slightly smaller. The difference between the two methods increased with the increase in the mean velocity of the gap flow and decreased with the increase in the gap ratio, which indicated that the larger the Reynolds number, the more difficult it was to measure the turbulent flow.

3.2. Boundary Layer Thickness

The boundary layer thicknesses of the gap flow for different conditions are shown in Figure 7. As can be seen from the figure, when the gap ratio was constant, as the mean velocity of the gap flow increased, the effect of additional shear stress in the turbulence of the fluid near the wall increased and the velocity gradient near the wall increased. The velocity distribution in the main flow area was more uniform and full, and thus, the thickness of the boundary layer decreased. When the mean velocity of the gap flow was constant, the boundary layer thickness decreased as the gap ratio increased, but the ratio of the boundary layer thickness to the gap half-height remained almost constant, which indicated that the velocity distributions of the gap flow were comparable in the range of gap ratios investigated in this study.

3.3. Wall Function

Using the wall friction velocities calculated in Section 3.1, the boundary layer data were nondimensionalized to obtain the velocity distributions in the boundary layer of the gap flow. The flow structure of the boundary layer of the gap flow also had obvious partitioning characteristics, as shown in Figure 8, where the boundary layer could be divided into an inner region and outer region along the normal direction, and the inner region could be divided into a viscous sub-layer, transition layer, and logarithmic layer. The dimensionless velocity distribution in the inner region was defined as the wall function.

3.3.1. Viscous Sub-Layer

In the region particularly close to the wall, the fluid was confined by the wall only as a viscous vortex diffusion under molecular viscous forces. Because the layer was so thin, the shear stress was considered to be invariant in the layer, and therefore, the velocity was linearly distributed. As seen in Figure 9, the velocity profiles were straight lines within $y$ < 0.08 mm; this region was called the viscous sub-layer. The wall function was expressed as

$$u^{+}=y^{+}$$

(11)

As shown in Figure 9, (a) demonstrates the velocity distributions in the viscous sub-layer of the gap flow boundary layer with different gap flow mean velocities when $\beta$ = 0.6/0.7/0.8, (b) demonstrates the velocity distributions in the viscous sub-layer of the gap flow boundary layer with different gap ratios when $u_m$ = 1 m/s/1.5 m/s/2 m/s, and (c) demonstrates the dimensionless velocity distributions in the viscous sub-layer of the gap flow boundary layer with different gap flow mean velocities and different gap ratios. As can be seen from Figure 9c, the dimensionless velocity profiles were independent of both the mean velocity of the gap flow and the gap ratio, and the velocity measurement points all fell in the vicinity of $u^{+}=y^{+}$ when $y^{+}<$ 5.5. From Figure 9a, it can be seen that with the increase in the mean velocity of the gap flow, the molecular viscous force increased, resulting in a greater slope of the velocity profile, i.e., the velocity gradient increased; for the same reason, Figure 9b shows that with the increase in the gap ratio, the velocity gradient also increased, which corresponds to Table 2.

3.3.2. Logarithmic Layer

For the outer region, it can be proposed that the wall acts to reduce the velocity from the maximum or centerline velocity to the velocity at the junction of the outer and inner regions. It is conventionally argued that the velocity scale for the outer region is also determined by the wall shear stress. Here, the dimensional velocity profile should behave as

$$\frac{u-u_c}{u_{*}}=g\left(\frac{y}{h/2}\right)$$

(12)

where $u_c$ is the centerline velocity, $u_{*}$ is the wall friction velocity, and the length scale in the outer region is the half-width of the gap $\left(\frac{h}{2}\right)$. Using this assumption, Millikan [27] proposed that there may be a region of overlap where both Equations (5) and (12) are simultaneously valid. The variables used in the inner region are $\frac{u}{u_{*}}$ and $\frac{y{u}_{*}}{\nu }$, and the variables used in the outer region are $\frac{u-u_c}{u_{*}}$ and $\frac{y}{h/2}$. Since the two methods, which use different parameters, correlate overlapping regions of the velocity profile, a relationship must exist between the two sets of parameters.

Equation (5) can be written as

$$\frac{u}{u_{*}}=f\left[\left(\frac{y}{h/2}\right)\cdot \left(\frac{\left(h/2\right)u_{*}}{\nu }\right)\right]$$

(13)

From Equation (12), we have

$$\frac{u}{u_{*}}=\frac{u_c}{u_{*}}+g\left(\frac{y}{h/2}\right)$$

(14)

A comparison of Equations (13) and (14) shows that the effect of the multiplicative factor $\frac{\left(h/2\right)u_{*}}{\nu }$ inside the function $f$ must be equivalent to the additive term $\frac{u_c}{u_{*}}$ outside the function $\mathrm{g}$. The logarithm is the only function with this property. Therefore, the overlap region is often referred to as the logarithmic region, where the velocity distribution satisfies

$$\left\{\begin{array}{c}\frac{u}{u_{*}}=Aln\left(\frac{y{u}_{*}}{\nu }\right)+B\\ \frac{u-u_c}{u_{*}}=Aln\left(\frac{y}{h/2}\right)+C\end{array}\right.$$

(15)

where the coefficients $A,B$, and $C$ are empirical constants; $A=\frac{1}{k}$; and$k$ is Kármán’s constant.

As shown in Figure 10, the experimental data under different conditions were fitted to obtain the coefficients of the logarithmic region wall function, that is, $A$ = 2.78, $B$ = 3.8, and $C$ = −0.9, corresponding to Kármán’s constant $k$ = 0.36. The maximum fitting errors of the two formulas were 2.58% and 3.9%, respectively, which indicate that the fitting function had high precision.

As seen in Figure 10a, the velocity profile satisfied a logarithmic law distribution when $y^{+}>$ 26.

Table 4. Thicknesses of logarithmic layer under different conditions.

		${\mathit{u}}_{\mathit{m}}$	1 m/s	1.5 m/s	2 m/s
	${∆\mathit{y}}^{+}∕\frac{{∆\mathit{y}}^{+}}{{\mathit{\delta }}^{+}}$
$\mathit{\beta }$
0.6			59/0.126	94/0.143	129/0.154
0.7			41/0.114	66/0.131	95/0.148
0.8			18/0.074	40/0.117	55/0.127

lists the thicknesses of the logarithmic layer under different conditions, as expressed using the dimensionless thickness ${∆y}^{+}$ and the ratio of the logarithmic layer thickness to the boundary layer thickness $\frac{{ ∆y}^{+}}{{\delta }^{+}}$. As can be seen from Table 4, as the mean velocity of the gap flow increased, the logarithmic layer thickness also increased, and the increase in the logarithmic layer thickness decreased with the increase in the gap ratio. However, the logarithmic layer thickness decreased as the gap ratio increased, and the decrease in the logarithmic layer thickness increased with the increase in the mean velocity of the gap flow. In addition, the percentage of the logarithmic layer increased with the increase in the mean velocity of the gap flow and decreased with the increase in the gap ratio. The specific data are shown in Table 4.

3.3.3. Transition Layer

The transition layer is located between the viscous sub-layer and the logarithmic layer, where the role of viscous forces begins to weaken and the role of Reynolds stresses becomes more prominent. Within this layer, the effects of viscous forces and Reynolds stresses are comparable, and the wall function differs considerably from that of the viscous sub-layer and the logarithmic layer. Rannie [28] and Tardu et al. [29] proposed using the hyperbolic tangent function to represent the relationship between $u^{+}$ and $y^{+}$ in the transition layer, as shown in Figure 11, where the expression is as follows:

$$u^{+}=\frac{1}{K}tanh\left(K{y}^{+}\right)$$

(16)

where $K$ is an empirical constant that is obtained using regression analysis based on experimental data under different conditions, as listed in

Table 5. Coefficients of the fitted curves under different conditions.

		${\mathit{u}}_{\mathit{m}}$	1 m/s	1.5 m/s	2 m/s
	K
$\mathit{\beta }$
0.6			0.0708	0.0695	0.0716
0.7			0.0717	0.0704	0.0711
0.8			0.0715	0.0713	0.0712

. The correlation coefficients of the fitted curves were all above 0.99, indicating that the functions were well fitted. All the constants were averaged to obtain the best-fitting relationship equation as follows:

$$u^{+}=\frac{1}{0.071}tanh\left(0.071y^{+}\right)$$

(17)

4. Discussion

In the past, when using hot-wire velocimetry and laser velocimetry for flow measurements, both the hot-wire probe and the laser control body could not be moved to the wall, that is, they could not be located exactly at the origin of the wall, and thus, the position of the wall could not be determined. It is common practice to use the logarithmic law formulation [30] of the velocity profile:

$$u^{+}=\frac{1}{k}ln\left(y^{+}\right)+B$$

(18)

or the full-wall law formulation [31] of the velocity profile:

$$y^{+}=u^{+}+e^{-kB}\left[e^{k{u}^{+}}-1-k{u}^{+}-\frac{{\left(k{u}^{+}\right)}^2}{2}-\frac{{\left(k{u}^{+}\right)}^3}{6}\right]$$

(19)

to determine the position of the wall; however, the values of the coefficients $\mathrm{k}$ and $\mathrm{B}$ have not yet been agreed upon. In this study, we proposed a new method to determine the position of the wall that is more accurate and simpler than the method using the velocity profile law. Figure 12 shows a photograph of the particles near the wall. As can be seen from the photo, in the vicinity of the wall, there are many groups of particles of the same size and the opposite shape. This was because the wall used in the experiment was made of Plexiglas, and the mirror function of Plexiglas allowed every particle to form an identical image on the other side of the wall. Therefore, the middle position of the particle and its image was the position of the wall.

From the experimental data found in this study, it was obtained that the range of the inner region of the gap flow boundary layer was $y<$ 0.18$\delta$ or $y<$ 0.13($h$/2), i.e., the influence of viscosity on the gap flow was within 0.18$\delta$ or 0.13($h$/2). The results obtained by different scholars for the ranges of the viscous influence on different types of boundary layers are shown in

Table 6. Ranges of viscous influence on different types of boundary layers.

Types of Boundary Layers	Ranges of Viscous Influence
Gap flow boundary layer	$0.18\mathsf{\delta }$$\mathrm{or} 0.13 (\mathrm{h}$/2)
Round pipe flow [32,33]	$0.1\mathrm{R}$$, 0.12\mathrm{R}$
Flat plate boundary layer [34,35]	$0.2\mathsf{\delta }$$, 0.3\mathsf{\delta }$
Open flow boundary layer [36,37,38,39,40]	$\left(0.1\text{–}0.25\right)\mathsf{\delta }$$, \left(0.15\text{–}0.5\right)\mathrm{h}$

. In round pipe flow and gap flow, the restriction of the radius or gap height inhibits the development of the boundary layer, which reduces the viscous influence range compared with the boundary layer on the flat plate in an infinite flow, whereas the free water surface in open flow can be varied arbitrarily due to the change in the displacement thickness of the boundary layer, and thus, its viscous influence range fluctuates greatly.

Although both the gap flow and the round pipe flow are pressurized flows, the differences in the wall shapes make the velocity distributions in the logarithmic regions different. Comparing the logarithmic region of the gap flow with the logarithmic region of the round pipe flow, as shown in Figure 13, the red solid and green solid lines in the figure are the logarithmic regions of round pipe flow measured by Zagarola [32] and Mckeon [33], respectively; the blue solid line is the DNS data of Kim [17]; and the dashed line is the result obtained by fitting our experimental data. Kim calculated the logarithmic region of the gap flow and found that Kármán’s constant $\mathrm{k}$ was 0.4 and the coefficient $B$ was 5, while Zagarol and Mckeon measured the logarithmic regions of the round pipe flow and obtained that the Kármán’s constant $k$values were 0.436 and 0.421, respectively, and the coefficient $B$ values were 6.13 and 5.6, respectively. On the whole, our experimental results were in good agreement with the DNS data obtained by Kim. However, when $y^{+}$ increased, the experimental data moved closer to the results of Zagarol and Mckeon. It can be seen that the difference between the gap flow and the round pipe flow was obvious when ${\mathrm{y}}^{+}$ was small, and the difference between the two gradually decreased as $y^{+}$ increased. Zagarol and Mckeon argued that the velocity distribution in the round pipe flow satisfied a power law rather than a logarithmic law when $y^{+}$ was small, and that the velocity distribution conformed to a logarithmic law only when${ y}^{+}$ was sufficiently large; however, no such law was found in the boundary layer of the gap flow in this study.

5. Conclusions

In this study, a series of experiments were carried out on the flow structure characteristics of the turbulent boundary layer of the gap flow based on PIV, and the following conclusions were obtained for Reynolds numbers of 16,587–56,870 and gap ratios of 0.6–0.8:

(1)

In this study, two methods were used to calculate the wall friction velocity, the results of the two methods were very similar, and the maximum relative error did not exceed 5%, which indicates that using PIV to measure the velocity profile in the viscous sub-layer to solve the wall friction velocity had good precision.

(2)

The boundary layer thickness of the gap flow was inversely proportional to both the mean velocity of the gap flow and the gap ratio, and the ratio of the boundary layer thickness to the gap half-height was constant for all the conditions investigated in this study.

(3)

The boundary layer data were nondimensionalized by using the wall friction velocity and the wall position to obtain the wall function of the boundary layer of the gap flow; it was expressed as

$$\left\{\begin{array}{c}{y}^{+}<5.5,|u^{+}=y^{+} \\ 5.5<y^{+}<26,|u^{+}=\frac{1}{0.071}tanh\left(0.071y^{+}\right)\\ y^{+}>26,|u^{+}=2.78ln\left(y^{+}\right)+3.8\end{array}\right.$$

(4)

In the viscous sub-layer, the velocity gradient was proportional to both the mean velocity of the gap flow and the gap ratio, and thus, the wall friction velocity was also proportional to the mean velocity of the gap flow and the gap ratio; the experimental data in the transition region satisfied the hyperbolic tangent function well; and the logarithmic region satisfied both the logarithmic law and the law of defects, and was therefore also known as the overlap region.

(5)

The thickness of the logarithmic region increased with the increase in the mean velocity of the gap flow and decreased with the increase in the gap ratio. The range of the inner region of the gap flow boundary layer was $y<$0.18$\delta$ or $y<$0.13($h$/2).

This study is of great significance for the study of gap flow theory. However, the static boundary is a special case in the category of moving boundaries, and thus, it is necessary to study the gap flow boundary layer under moving boundary conditions in the future and compare the results with those of the static boundary.