Effect of d -type rib roughness on the turbulent structure of side wall boundary layer for wave-current combined flow

Abstract

An experimental study has been carried out in a laboratory flume to characterize the turbulence structure and turbulence anisotropy in the boundary layer over smooth and rough side walls for both current alone and wave-current combined flow situations. The rough side wall of the flume comprises a train of circular ribs (diameter, k) attached vertically maintaining uniform spacing p along the streamwise direction. The experiments are performed for smooth surface and rough (ribbed) surfaces with p/k = 2, 3, and 4 to reproduce different cases of d-type rib roughness. The effect of wave-current interaction has been investigated by superposing waves of two different frequencies. Time series data of three velocity components are obtained using Acoustic Doppler Velocimeter. At the near wall region, roughness with higher p/k value enhances the level of turbulent intensity and Reynolds stress significantly. In a channel with smooth side wall, the wave-current combined flow produces lesser turbulence intensity than the current alone flow near the wall. However, for a ribbed wall case, the effect is completely opposite that is, wave-current interacting flow induces higher intensities compared to the reference current alone flow. Substantial decline in the turbulent length scales at the near wall region are observed for ribbed walls, which reveals the strong effect of roughness elements on the turbulent structure. Superposition of wave reduces the length scales even more for both smooth and rough wall cases. As the spacing between two ribs (p/k ratio) increases, the energy dissipation rate increases. The analysis of anisotropy invariant map demonstrates a reduction of anisotropy in the vicinity of ribbed wall compared to that for a smooth wall. For wave-current combined flow, the anisotropy invariant data of Reynolds stress tensor varies dramatically within the boundary of map, reflecting significant changes in the state of turbulence.

Keywords

Wave-current combined flow side wall boundary layer turbulent length scales Reynolds stress anisotropy Anisotropy invariant map

Introduction

Many fluid engineering systems deal with turbulent flows induced by uneven surfaces. The surface roughness essentially increases both the flow resistance and the rate of momentum transfer. To modulate the turbulence structure in the flow, surfaces are often artificially roughened. Since Nikuradse’s¹ seminal work on flow through rough pipes, turbulent flow over uneven surfaces has been investigated extensively due to its technological importance. Studies on the turbulence characteristics of coexisting wave and current flow has been scarce despite the fact that such wave-imposed current flows are of great importance for the understanding of flow resistance and sediment entrainment in the coastal environments. The combination of wave and current flows govern many physical processes of interest to the engineers and researchers working in coastal environments. Grant and Madsen² carried out a theoretical analysis of wave-current combined flow over a rough surface boundary and reported an increase in apparent bed roughness and shear stress when waves were superimposed on the current. Later Kemp and Simons³ performed an experimental study in a laboratory channel for wave-current combined flow over smooth and rough boundaries and reported a considerable increase in bed shear stress for rough boundary when waves are superimposed on the current flow. Moreover, Kemp and Simons⁴ documented a significant reduction in sidewall boundary layer thickness and mean motion in the vicinity of the sidewall for wave-current combined flow. Further the results indicated that a redistribution of flow occurs across the channel for such wave superimposed flows. Thus, the flow structure of wave current combined flows has been a subject of great interest to numerous academicians and researchers. Yu et al.⁵ conducted a study to resolve the turbulent scales for oscillatory boundary layer over rough wall covered with two-dimensional roughness elements. They reported that the dynamics depends on both the shape and size of the roughness elements and their relative positions that is, pitch-to-height ratio. Yu et al.⁵ also examined the anisotropy of turbulence through anisotropy invariant map (AIM) analysis for rough wall oscillatory boundary layer and found that turbulence becomes more isotropic when moves toward the outer boundary layer.

In this paper the turbulence characteristics are presented for wave-current combined flow over d-type rib roughness on the side wall of a channel based on laboratory flume experiments. One of the important parameters for the characterization of turbulence is the turbulent length scales which are based on measured velocity time series. The invariant analysis of Reynolds stress tensor⁶ is also presented in the present study, to extract information about the anisotropy of large-scale motion.⁷ In view of applying turbulence closure models in RANS based CFD simulations, it is of great importance to have some prior knowledge of expected anisotropy of the flow field under exploration. Shafi and Antonia⁸ suggested that the knowledge of this anisotropy is of much help in the determination of the sensitivity of turbulence structure to different boundary conditions. Particular interest was paid to the sidewall near boundary turbulent flow within the roughness sublayer, with the concept of wall similarity hypothesis that turbulent motion outside the roughness sublayer are almost independent of the wall boundary condition.^9–11 Krogstad and Antonia¹² indicated the differences of large-scale structure between the smooth and rough walls in view of anisotropy of Reynolds stress between the smooth and rough wall flows.

Various roughness elements such as sand grains, gravels, spheres, cubes, wire mesh and two-dimensional ribs (circular, square) attached to the boundary wall surfaces were investigated in the past to model surface roughness. Understanding the flow characteristics over these roughness elements received a lot of attention due to its application in numerous areas of engineering and environmental sciences. According to Perry et al.,¹³ the rib roughness is classified as d-type and k-type based on the pitch-to-roughness height ratio (p/k): p/k < 5 considered as d-type, p/k > 5 considered as k-type and p/k = 5 as intermediate roughness. Ribs roughness are employed in many engineering applications to augment heat transfer rates and also often used to study the effects of surface roughness on momentum transport.¹⁴ Understanding the impact of roughness on the transport and mixing of contaminants such as wastewater and oil discharged into river systems by industries or natural disasters is critical from an environmental point of view.¹⁵ To improve engineering design practice, a thorough understanding of roughness effects on mean motion and turbulence characteristics is required. Turbulent flow over uneven surfaces are far more complicated than flow over flat ones. The interaction of the flow with the protrusion of the roughness elements attached to the boundary wall, both in terms of mean velocity and turbulent quantities is complex. Further when the turbulence generated by the wave-current combined flow interacts with the turbulence caused by roughness elements, the flow becomes more complex.

Numerous experimental and numerical investigations have facilitated the acquisition of information on the complex behavior of wave-current interaction flow over smooth surface and various rough surface boundaries. Several studies on the interaction of wave and current flow over smooth boundary have been carried out experimentally^3,16–19 and numerically^5,20–22 to examine the mean velocities, turbulence intensities and Reynolds stresses. Further equal importance over rough boundary has been given from past. Mostly the studies were focused on the bottom roughness such as ripple bed,^23–25 series of square rib elements spanning the width of the channel bed surface,²⁶ 5 mm high triangular wooden strips stuck on the bed across the channel width with the spacing between two consecutive strips about 18 mm which corresponded to p/k < 5 known as d-type roughness.³ Similarly, Barman et al.²⁷ used chain of hemispherical ribs with p/k = 2, 4, and 8 as the bottom roughness. Kemp and Simons³ showed that the near-bed velocity decreased over a rough surface, whereas the near-bed turbulence intensity increased in the presence of waves. Kemp and Simons⁴ investigated the sidewall boundary layer thickness in addition to the bottom boundary layer and reported that sidewall boundary layer thickness dramatically reduced by approximately 4 times that of current alone flow for the wave current combined flow. The overall result of the study of Kemp and Simons^3,4 indicated a redistribution of flow across the channel for wave current combined flow. It has been an important subject to researchers with relevance to the pipe flow, canal and river flow and river bank erosion process. However, vertical bank or sidewall for canals and rivers comprises of different roughness scales that introduce additional turbulence scales at the bottom boundary layer turbulence. Hansda and Debnath²⁸ carried out an experimental study to look into the distribution of mean velocity, turbulence intensity, Reynolds shear stress and streamwise integral length scales over a sand-grain rough and granule rough sidewall boundary under the wave-against current flow. Investigation of turbulence structures and eddy scales close to the sand-grain and granule rough sidewall surfaces due to interaction with wave-current combined flows were presented in Hansda et al.²⁹ By varying the submerged vegetation on a vertical wall, velocity and the turbulence structure was studied for waves, current alone and wave-current combined flow by Chen et al.³⁰ Throughout the water column, the obtained turbulent kinetic energy was relatively larger in magnitude for wave-current combined flow than the current alone flow. The riverbanks may have varying combination of submerged and non-submerged vegetation.³¹ These vegetations attribute to varying degrees of roughness in the bank wall³² and the roughness influenced the turbulent kinetic energy³³ Further Czarnomski et al.³⁴ reported that the turbulence intensities near the bank wall influenced by the bank vegetation roughness. Yu et al.³⁵ analyzed the boundary layer dynamics and bottom friction in wave-current combined flows over large roughness elements and showed that waves enhance the drag force on the current flow and time averaged Reynolds stress at the top of the roughness.

In the present paper, flow and turbulence structure under the influence of d-type rib roughness (p/k < 5) attached to the side wall of the channel are extensively investigated for both current alone flow and wave-current combined flow. By varying the spacing (p) between successive circular ribs, the equivalent sand-grain roughness, k_s,¹ has been altered, and thus, the turbulence characteristics of the rough boundary layer has been amended without changing the roughness height, k. Cui et al.³⁶ showed that within the cavity of two successive rib elements, velocity, Reynolds stress and turbulent kinetic energy are very small, and each of these variables reaches a maximum near the rib height. Traditionally, most studies in this field portray the distribution of pertinent parameters such as mean velocity, turbulent intensity, Reynolds stress, higher order moments, eddy viscosity, mixing length, turbulent kinetic energy over both d-type and k-type rib roughness for turbulent current-only flow. Thus, there is indeed a dearth of investigation for wave-current interacting flow although it appears in a wide spectrum of real-life applications (e.g. environmental flows). In view of that, this paper undertakes a details study of turbulent characteristics of the side wall boundary layer over d-type rib roughness attached to the sidewall of the channel with a special emphasis on the wave-current combined flow. Here, two different wave frequencies (f = 0.4 and 0.8 Hz) are superimposed on the main flow. In order to explore the effects of the variation of p/k and wave-current interaction on the turbulent length scales (largest integral length scale, intermediate Taylor scale and smallest Kolmogorov scale) and anisotropy of Reynolds stress, time series data of velocity components are obtained using Acoustic Doppler Velocimeter (ADV). Turbulence characterization is carried out through the estimation of turbulent length scales in which both the largest scales comprising the large-scale energy production and the smallest scales comprising the energy of dissipative scales are examined. One of the most interesting findings of the present study is that the superposition of wave reduces the integral length scale (∼ largest eddy size) for both smooth and rough wall cases. With an increased wave frequency (f = 0.8 Hz case) the largest eddy size is reduced even further. The estimates of smallest or Kolmogorov length scale demonstrate similar effects of wall roughness and wave. The role of d-type roughness on energy dissipation rate is thoroughly investigated, too. The anisotropy invariant map (AIM) of Reynolds stress tensor is carefully computed as understanding the anisotropic structure of the turbulent flow field is of considerable interest for RANS based computational analysis. AIMs of current alone flow and wave-current combined flow for various p/k wall roughness (d-type) are compared with the respective AIM data obtained for smooth side wall cases. Results show intriguing diversity in the state of turbulence, reflecting intricate interaction of d-type wall roughness and superimposed surface waves.

Experimental details

Water flume and experiment setup

Experiments were performed in an 18 -m-long open channel flume situated at the fluid mechanics and hydraulics laboratory, Indian Institute of Engineering Science and Technology (IIEST), Shibpur, India. The width and depth of the channel were 0.6 and 0.9 m, respectively. The flume wall at the test region was made of transparent Perspex to facilitate proper visualization of the flow. The “rough side wall” was fabricated by attaching a train of circular wooden rods (ribs) to it, as shown in Figure 1(b). The rib diameter (k), which essentially represents the roughness height, was kept constant (k = 15 mm) during the investigation. The roughness characteristics of the side wall was altered by varying the spacing between two successive ribs as p = 30, 45, and 60 mm which yielded pitch-to-roughness height ratio, p/k = 2, 3, and 4, respectively (see Figure 1(c)). All three p/k values considered here conform to the classical definition of d-type roughness.¹³

Figure 1.

(a) Schematic of open channel (plan view) showing rough side wall, measurement location and coordinate system (not to scale), (b) photograph of rough side wall and ADV, and (c) sketch of p/k = 2, 3, and 4 and data collection position (x_c).

To achieve the d-type rough side wall, the ribs were glued onto the side wall, according to the chosen spacing (p) value, vertically that spanned the height and entire length of the channel wall (Figure 1(b)). Uniform flow was maintained by using series of wire mesh perforated baffle walls at entrance of the channel. The mean (depth average) velocity (U) of current alone (CA) flow and water depth (h) were maintained as 34.6 cm/s and 20 cm for all experimental runs. Thus, in the present experimental program, all the experimental runs were carried out for the Reynolds number $(Re = Uh / Uh ν) ν) 6.92 \times 10^{4}$ where $ν$ is the kinematic viscosity of water. The uncertainty was estimated as 2.4% and 1.5% for depth average velocity and water depth respectively. To produce the wave-current combined (WCC), a cylindrical plunger-type wavemaker fitted with a slider-crank mechanism installed at the inlet of the flume channel was operated. The wave maker consisted of a cylindrical (20 cm diameter) plunger attached with two connecting rods, gear mechanism and a motor. The rotational speed (rpm) of the motor was controlled by a DC variac. The reciprocating motion of the plunger generated the desired wave frequency by controlling the rpm of motor. As a consequence, the progressive waves of desired frequency were produced with certain wave height (H) and wave length ( $λ$ ) with the combination of current flow. For the current alone (CA) flow the plunger was kept above the water surface. In the present experiment two different wave frequencies (f) 0.4 and 0.8 Hz are superimposed into the current alone flow to obtain two different wave-current combined flow conditions.

For the purpose of velocity data collection in the wave current combined flow the 16 MHz micro acoustic Doppler velocimeter (ADV) was used in the experiment. The ADV was mounted in a carriage which allowed traversing in the x, y and z direction corresponds to streamwise, transverse and vertical direction respectively of the flume channel (see Figure 1(b)). The ADV system was equipped with a transmitter and three receivers. The intersection of the receiver axes designates the location of sampling volume and it was located 5 cm away from the transmitter. The sampling volume of ADV was 0.09 cm³ with a cylindrical shape. The velocity data of the flow field were collected by ADV at 40 Hz sampling frequency rate with specified factory calibration as ±1% of the measured velocity (i.e. accuracy of ±1 cm/s is on the measured velocity of 100 cm/s).

Prior to the main measurement in the test region, a preliminary experiment was performed for current alone flow to check flow uniformity in the test region. For this purpose, two horizontal component of velocity u and v correspond to streamwise and transverse direction respectively were measured at 20 locations in x-y plane of the channel at 10 cm (z/h = 0.5) above the bottom within the test region (10 m < x <13 m; 0.005 m < y < 0.3 m). The positions in the x directions were 10, 11, 12, and 13 m from inlet of the channel and in the y direction were 0.005, 0.5, 0.1, 0.2, and 0.3 m from the wall. Figure 2 shows the mean velocity $\bar{u}$ , $\bar{v}$ and the standard deviation of $\bar{u}$ and $\bar{v}$ for current alone case. It is pertinent to mention here that for a uniform flow for the current alone case $\bar{u}$ would be nearly constant and almost zero for $\bar{v}$ in the region.³⁷ The Figure 2(a) and (b) exhibits almost constant value of $\bar{u}$ and almost zero of $\bar{v}$ in the test region. Figure 2(b) shows slight reduced velocity near the side wall (y = 0.005 m from the wall) and it probably due to the boundary wall effect.

Figure 2.

Current alone flow in preliminary experiment at z/h = 0.5: (a) x-direction (y = 0.3 m) and (b) y-direction (x = 12 m).

Also, the stability of the incoming uniform flow in the channel was verified by the equation (1),^38,39 derived by Vedernikov⁴⁰ using an approximation of the Saint-Venant equation.

V_{n} = χ ψ Fr

(1)

Where $V_{n}$ = Vedernikov number, $χ$ = Exponent of the hydraulic radius and its value is 1/2 and 2/3 for the Darcy-Weisbach and Manning formulas respectively, $ψ$ = shape factor of the channel section and $Fr$ = Froude number. The shape factor and Froude number determined by

ψ = 1 - R_{h} \frac{dP}{dA}

(2)

Fr = \frac{U}{\sqrt{gD \cos θ / α}}

(3)

Where $R_{h}$ = hydraulic radius, P = wetted perimeter, A = water area, U = cross section averaged mean velocity, $D = A / A W W$ = hydraulic depth, where W is the flow width; θ = slope angle of the channel bottom, g = gravitational acceleration. $α$ = energy coefficient and for simplicity it is assumed to unity.³⁹ The $V_{n} > 1$ represents the unstable flow, $V_{n} < 1$ represents the stable flow and $V_{n} = 1$ is considered as neutral stability. In the present experimental study, the shape factor (equation (2)), $ψ$ = 0.6 for the open channel section (width = 60 cm, water depth = 20 cm) and the Froude number (equation (3)), Fr = 0.24 (θ = 0.00025, D = W = 20 cm for rectangular channel section). Hence the $V_{n}$ = 0.09 obtained which conforms the flow stable criteria ( $V_{n} < 1$ ). Therefore, these results conclude that the current alone flow for the present experiment background conditions is uniform and stable in the test region.

The wave generation by the installed plunger type wavemaker was validated with the second order Stokes wave theory for pure wave case without current. Therefore, an experiment was performed in the flume for wave alone (WA) case using the experiment methodology as reported in Sarkar et al.⁴¹ using the same experimental set-up. The water depth was kept at 20 cm and the oscillation time period of plunger of the wavemaker was set as T = 1.23 s ( $f \approx 0.8$ Hz). The measured time series data of free surface elevation ( $η$ ) and streamwise direction velocity (u) in the experiment were compared with the theoretical profile given by the second order Stokes wave theory. The wavelength ( $λ$ ) and wave height ( $H$ ) found as 72 and 4.4 cm respectively hence the wave steepness ( $H / H λ λ$ ) is 0.061 for the present wave alone case in the experiment.

The velocity potential ( $φ$ ) of second-order Stokes wave theory and the free surface profile ( $η_{N}$ ) are given as follows.⁴²

\begin{matrix} φ (x, z, t) = \frac{Hg}{2 ω} \frac{\cosh K (z + h) \sin (Kx - ω t)}{\cosh (Kh)} \\ + \frac{3}{32} \frac{ω H^{2}}{\sinh^{4} (Kh)} \cosh (2 K (z + h)) \sin (2 (Kx - ω t)) \end{matrix}

(4)

\begin{matrix} η_{N} = \frac{H}{2} \cos (Kx - ω t) \\ + \frac{K H^{2}}{16} \frac{\cosh (Kh)}{\sinh^{3} (Kh)} (2 + \cosh 2 Kh) \cos 2 (Kx - ω t) \end{matrix}

(5)

Here H is the wave height and $ω (= 2 π / 2 π T) T)$ is the angular frequency of the wave, and $K (= 2 π / 2 π λ) λ)$ is the wavenumber. The local velocity in the longitudinal direction ( $u_{N}$ ) under the second order Stoke wave theory is given in equation (6). This is obtained by partially differentiating the potential function ( $φ$ ) (equation (4)) with respect to x.

\begin{matrix} u_{N} = \frac{Hg}{2 ω} \frac{1}{\cosh (Kh)} K \cosh (K (z + h)) \cos (Kx - ω t) \\ + \frac{3}{32} \frac{ω H^{2}}{\sinh^{4} (Kh)} 2 K \cosh (2 K (z + h)) \cos (2 (Kx - ω t)) \end{matrix}

(6)

The second order Stokes wave theory is applicable at intermediate depths ( $0.05 \leq h / h λ \leq 0.5 λ \leq 0.5$ ) and deep-water depths ( $h / h λ λ \geq 0.5$ ). The applicability of this theory must satisfy the Ursell number ( $U_{r} < 40$ ) criteria.^43,44 The Ursell number is based on the water depth (h), the wavelength ( $λ$ ) and wave height (H) and it is determined by the equation (7).

U_{r} = \frac{H λ^{2}}{h^{3}}

(7)

Here in the present experiment for wave alone case, the Ursell number is approximately equal to 2.85 and follows the Ursell number criteria to apply second order Stokes wave theory.

Using the equation (5), for the experimental wave characteristics, $T$ = 1.23 s, $H$ = 4.4 cm, and $λ$ = 72 cm; and analytically the $η_{N}$ was determined as the timeseries data. In this regard the good approximation results of $η_{N}$ depend on the time steps. Havn⁴⁵ and Silva et al.⁴⁶ showed that the time steps of $T / T 100 100$ provide a good approximation. Similarly Finnegan and Goggins⁴⁷ proposed the time steps of $T / T 50 50$ and Marques Machado et al.⁴⁸ performed the simulation with time steps 0.005 s ( $\approx T / \approx T 400 400$ ) and 0.02 s ( $\approx T / \approx T 100 100$ ) and concluded that the optimum time steps may be the corresponding to the $T / T 200 200$ . In equation (5) the time steps 0.004 s ( $\approx T / T 308 308$ ) is considered to determine the analytical results of surface elevation ( $η_{N}$ ) of the wave because the measuring instrument (wave-gauge) used in the experiment acquire the surface elevation ( $η_{E}$ ) data with time steps of 0.004 s. Figure 3 shows the comparison of surface elevation between the results of analytical ( $η_{E}$ ) and experimental ( $η_{E}$ ). From the Figure 3(a) it is observed that the waves produced by the wavemaker almost coincide with that predicted by second order Stoke wave theory. However the small differences between results can be observed by close inspection and may be due to limitations in experiments or the second order Stoke wave theory. To understand the consistency of surface elevation profile throughout the entire length of experiment, the ensemble average of wave cycles of surface elevation profile is compared with the analytical result. The comparison plot between experimental and analytical result of average of wave cycles of surface elevation shown in Figure 3(b). It can be observed in Figure 3(b) that relative error of wave heights (i.e. difference between the elevations of a crest and neighboring trough⁴⁹) is about 8.6% between experimental and analytical.

Figure 3.

(a) Comparison of surface elevation between experimental and analytical (Stoke second order equation) at wave steepness, H/λ = 0.061, (b) comparison of ensemble average wave cycles of surface elevation profile between experimental and analytical.

Using the equation (6), the longitudinal direction velocity ( $u_{N}$ ) is determined analytically applying the second order Stoke wave theory for the same wave characteristics as in the experiment, $T$ = 1.23 s, $H$ = 4.4 cm and $λ$ = 72 cm. In the experiment, the velocity u for wave alone case was measured by the ADV at a sampling rate of 40 Hz, 10 cm below the water surface (i.e. z = −10 cm). These timeseries data of u were compared with the analytically determined the u-velocity ( $u_{N}$ ) timeseries profile. The comparison plot of experimental and analytical result is shown in Figure 4. The Figure 4 indicate that the acquired timeseries data of velocity u by the ADV are in good agreement with the analytical result ( $u_{N}$ ) determined using the second order Stoke wave theory.

Figure 4.

Comparison of longitudinal velocity between analytical and experimental results for wave steepness, H/λ = 0.061 at z = −10 cm.

For the detailed experimental study on the turbulence structure on the side wall boundary layer due to the effect of d-type rib roughness for the wave-current combined flow two types of wave-current combined flow situations were considered and were designated as as WCC_0.4 and WCC_0.8. The WCC_0.4 and WCC_0.8 cases correspond to surface wave frequency f = 0.4 and 0.8 Hz superimposed respectively into the current alone flow. The wave characteristics for each type of wave-current combined flows are given in Table 1.

Table 1.

Wave characteristics at test section.

	CA	WCC_0.4	WCC_0.8
Water depth, h (cm)	20	20	20
Wave frequency, f (Hz)		0.4	0.8
Wavelength, λ (cm)		98	86
Wave height, H (cm)		2.1	2.7
Wave steepness, H/λ		0.021	0.031
Ratio of λ/H		46.7	31.8

The wavelength (λ) and wave height (H) of the wave-current combined flow are estimated from the captured images of progressive waves. To record the photographs of the progressive waves, a digital camera was focused perpendicular to the flume’s Perspex side wall. The photos had a resolution of 1280 × 720 DPI. The average wavelength and waveheight are calculated by first converting each image’s pixel unit to the metric unit (1 pixel = 0.026458 cm) and then averaging the results. The uncertainty of the computed averaged wavelength and wave height for both wave frequency cases are within 2%. The calculated wave steepness for WCC_0.4 and WCC_0.8 cases are 0.021 and 0.031, respectively (Table 1), and these values are much less than the limiting criteria (0.14) of wave breaking.^50,51 The ratios, λ/H = 46.7 and 31.8, obtained for WCC_0.4 and WCC_0.8 respectively, are well within the range of 4–56 proposed by Goda⁵² for waves in natural settings. It is worthwhile to distinguish the wave regions based on the ratio of water depth and wavelength (h/λ) as shallow water, intermediate depth and deep-water wave regions.⁴² The present wave cases represented intermediate ( $0.05 \leq h / h λ \leq 0.5 λ \leq 0.5$ ) depth wave category.

Measurements and methodology

The three components of velocity u, v and w corresponds to x, y and z direction were measured by the side looking 16 MHz micro-acoustic Doppler velocimeter (ADV). The three receivers of ADV system records simultaneously velocity component, signal strength, signal-to-noise ratio (SNR) and correlation value. The SNR and correlation values are used primarily to determine the quality and accuracy of the velocity data.⁵³ The minimum values of SNR and correlation are usually 15 dB and 70% that is required to record signals for turbulence characterization purposes.⁵⁴ Hence, the values of SNR and correlation lower than 15 dB and 70% are considered as poor-quality data which may lead to wrong interpretation in results. Also, in the ADV recorded velocity signals, there may be spikes caused by aliasing the Doppler signal and these spikes create ambiguity in the record.^55,56 Thus, the removable of poor-quality data and spikes from the record data is required to show the reliability of the quantitative results. The phase space threshold despiking developed by Goring and Nikora⁵⁶ is used to remove spikes from the 3-min recorded data. Also, to remove the plausible noise, the raw ADV data are processed by high pass filter with low cutoff signal-to-noise ratio (<15 dB) and correlation samples (<70%) and this is implemented by Win-ADV software. To visualize the consequence of filter process, an example of a record data of ADV for current alone flow situation shown in Figure 5.

Figure 5.

Sample record dataset of ADV used for filter test (a) signal-to-noise ratio data (b) overlapped plot of raw (unfiltered) data and filtered data of u velocity, (c) signal-to-noise ratio for small length of time 5 s and (d) overlapped plot of raw and filtered data for small length of time 5 s.

Figure 5(b) shows the comparison plot between raw ADV data and filtered data of u and Figure 5(a) shows their corresponding SNR value for length of 120 s. In the Figure 5(a) symbols (+) represents the SNR values and the dash-line (red color) is the low cut-off (15 dB) line. It means below the dash-line the SNR values and corresponding velocity (u) data are poor quality data, hence this data must be filtered from the record. To compare the filtered and raw data, the filtered timeseries data is overlapped with the raw data as shown in Figure 5(b). It is observed from the Figure 5(b) that the black spots in the overlapped timeseries data are actually eliminated data those have low SNR value (<15 dB) in raw timeseries data. For clarity of visualization a small length 5 s timeseries data are plotted in Figure 5(c) for SNR value and Figure 5(d) for velocity for comparison between raw and filtered data. In the present study to characterize the turbulence structure, the analysis was performed on the filtered data set. The velocity components u, v and w are decomposed into its mean and fluctuation parts as

u = \bar{u} + u'; v = \bar{v} + v'; w = \bar{w} + w'

(8)

Here $\bar{u}, \bar{v}$ , and $\bar{w}$ are time-averaged (mean) velocities, and $u', v'$ , and $w'$ are corresponding velocity fluctuations, respectively. For flow under combined wave-current interaction, apart from the mean velocity and turbulent fluctuation, an additional term namely wave-induced velocity appears in the decomposition of velocity.¹⁹ Thus, equation (8) becomes

u = \bar{u} + \tilde{u} + u'; v = \bar{v} + \tilde{v} + v'; w = \bar{w} + \tilde{w} + w'

(9)

Here tilde (∼) sign denotes wave-induced velocity. Since the wave-induced velocity is periodic in nature, it can be calculated from the phase averaged velocities¹⁹ as

\tilde{u} = < u > - \bar{u}; \tilde{v} = < v > - \bar{v}; \tilde{w} = < w > - \bar{w}

(10)

where the angle bracket (< >) represents phase averaged velocities. Turbulence intensities and Reynolds shear stresses are estimated from the turbulent velocity fluctuations. Accordingly, the streamwise (I_u), transverse (I_v) and vertical (I_w) turbulent intensities and the Reynolds shear stress (R_uv) in the x-y plane are expressed as

I_{u} = \sqrt{\bar{{u'}^{2}}}; I_{v} = \sqrt{\bar{{v'}^{2}}}; I_{w} = \sqrt{\bar{{w'}^{2}}}; R_{uv} = \bar{u' v'}

(11)

The details of all experimental test conditions and boundary layer parameters are summarized in Table 2. In the forthcoming discussion as well as in Table 2, SM, R1, R2, and R3 refer to the test case with smooth side wall, the cases of rough side walls with p/k = 2, p/k = 3, and p/k = 4, respectively. The mean velocity and the turbulent quantities were measured carefully over smooth side wall (SM) and rough/ribbed side walls (R1, R2, and R3) for current alone flow (CA) and wave-current combined flows (WCC_0.4 and WCC_0.8). All measurements were carried out along the transverse direction (perpendicular to the side wall) midway (x_c) between two successive ribs for p/k = 2, 3, and 4 (see Figure 1(c)). The measurement position (x_c) was located beyond the distance x = 9.5 m from flume inlet, where the flow was verified to be fully developed. The mean velocity profiles were measured along the transverse direction from the closest possible position (y/B = 0.015, where y = normal distance from the wall and B = half of channel width) of side wall to center line of channel at three different heights (h1, h2, and h3) of water for p/k = 2, 3, and 4. The three different heights h1, h2, and h3 are defined as z/h = 0.25, z/h = 0.5 and z/h = 0.75 respectively, where z = height from bottom bed and h = water depth. In this paper, the analysis is performed considering measured data at height h2 (z/h = 0.5) which we defined as mid-depth data.

Table 2.

Test conditions and boundary layer parameters.

Side wall surface	Test	p/k	z/h	Ue (cm/s)	U_τ (cm/s)	Π	ΔU⁺	k_s ⁺	k_s (cm)
				CA
Smooth	SM_h2		0.5	31.4	0.7	0.17
Rough	R1_h2	2	0.5	31.6	1.88	0.27	3.2	15.7	0.08
Rough	R2_h2	3	0.5	31.9	1.8	0.34	5	33	0.18
Rough	R3_h2	4	0.5	31.8	1.9	0.37	7	75	0.39
				WCC_0.4
Smooth	SM_h2		0.5	30.8	0.67	0.26
Rough	R1_h2	2	0.5	30.9	1.87	0.16	3.5	17.8	0.09
Rough	R2_h2	3	0.5	31.3	1.68	0.18	6	49.6	0.29
Rough	R3_h2	4	0.5	30.2	1.7	0.15	9	170	1
				WCC_0.8
Smooth	SM_h2		0.5	30	0.69	0.22
Rough	R1_h2	2	0.5	31.6	1.9	0.14	4	22	0.11
Rough	R2_h2	3	0.5	32.6	1.6	0.15	6.5	61	0.38
Rough	R3_h2	4	0.5	31	1.7	0.17	10.2	279	1.6

The shear velocity, U_τ for smooth wall surface is determined using the Clauser chart method⁵⁷ assuming the velocity profile follows the logarithmic form (equation (12)) in the overlap region of the boundary layer.⁵⁸

\frac{U (y)}{U_{τ}} = \frac{1}{κ} \ln (\frac{y U_{τ}}{ν}) + C

(12)

Here, $U (y)$ is the mean velocity at y (normal distance from the wall), ν = kinematic viscosity, $κ$ = 0.41 is the von Karman constant and C = 5 is the log-law intercept for smooth wall.⁵⁹ For the rough wall surface the U_τ is determined following the profile matching technique.^15,60,61 The profile is mean velocity defect profile which describes the logarithmic region of the boundary layer. The form of the profile is given below⁶⁰

\frac{U (y)}{U_{e}} = 1 + \frac{U_{τ}}{κ U_{e}} [\ln (\frac{y + y_{0}}{δ}) - (1 + 6 Π) {1 - {(\frac{y + y_{0}}{δ})}^{2}} + (1 + 4 Π) {1 - {(\frac{y + y_{0}}{δ})}^{3}}]

(13)

where $U_{e}$ = freestream velocity, $y_{0}$ = the location of virtual origin, δ = boundary layer thickness and Π = Coles wake parameter. The values of $U_{τ}, y_{0}$ and Π are optimized using a least-squares optimization method⁶⁰ to fit the equation (13) to the experimental data with reasonable agreement. Following the same procedure for each data set of velocity profile for rib roughness (p/k = 2, 3, and 4) the $U_{τ}$ and Π are determined and are given in Table 2. In Table 2 the data are presented for the h2 (z/h = 0.5) case that is, mid-depth of water for all experimental runs. The Π value significantly increased for ribbed wall surfaces when compared with smooth wall surface for current alone flow. With the increased of spacing between two ribs for d-type rib roughness the Π value increased. In the present case Π = 0.27, 0.34, and 0.37 for p/k = 2, 3, and 4 respectively are obtained. Similar results are reported in the experimental study for various rough surfaces^15,60,62 showing that the strength of wake component is greater for rough surfaces than the smooth surface. For wave-current combined flows this wake component, Π value over rib roughness boundary are found comparatively less than the value observed for current alone flow. This is because in the wave-current combined flow, the lee-wake vortex is destroyed over the rib crest.²⁴

The mean velocity profile in inner coordinates (Y⁺ vs U⁺) for smooth and rough boundary wall surfaces at h2 (SM_h2, R1_h2, R2_h2, and R3_h2) for CA, WCC_0.4 and WCC_0.8 flow cases are shown in Figure 6. In this figure (Figure 6), $Y^{+} = y U_{τ} / y U_{τ} ν ν$ and $U^{+} = U / U U_{τ} U_{τ}$ . The downward shift of mean velocity profiles for rough boundary wall surfaces indicates the roughness effect on mean velocity profiles and the magnitude of shift is known as roughness function (ΔU⁺). The decrease of velocity is strongly related to the friction Reynolds number and the geometrical properties of roughness such as roughness height, density and shape parameters.⁶³ The Table 2 reveals that the ΔU⁺ increases with the increase of p/k ratio for d-type rib roughness for flow cases CA, WCC_0.4 and WCC_0.8 presented herein. This is probably due to increase of spacing between the ribs that influences the roughness effect in the boundary layer and results in momentum deficit.⁶⁴ The dimensionless equivalent sand-grain roughness height ( $k_{s}^{+}$ ) is estimated using the following equation.^61,64

Δ U^{+} = \frac{1}{κ} \ln (k_{s}^{+}) - 3.5

(14)

Here $k_{s}^{+} = k_{s} U_{τ} / k_{s} U_{τ} ν ν$ , $k_{s}$ = a typical roughness scale known as equivalent sand-grain roughness height. For any roughness pattern, the value of roughness length scale, $k_{s}^{+}$ represents the size of uniformly packed sand grains from the experiments of Nikuradse¹ that produces the same frictional resistance as the surface of interest in the fully rough regime.⁶³ According to Flack and Schultz,⁶⁴ $k_{s}^{+}$ is more of a hydraulic property than a geo metrical scale of the surface that might be measured. Once the mean velocity profile is known, its value can be determined. The estimated $k_{s}^{+}$ are given in Table 2. The value of $k_{s}^{+}$ increases with the increase of p/k ratio for d-type rib roughness in current alone flow. When waves are added to the current alone flow the value of $k_{s}^{+}$ further increases. From Tables 1 and 2 it can be observed that wave current combined flow situation with higher wave height (WCC_0.8) exhibits relatively large roughness scale ( $k_{s}^{+}$ ) than the smaller wave height (WCC_0.4) for the same roughness height. However, in this regard more experimental study with variety of wave heights is required. Within the d-type rib roughness, the increase of p/k ratio increases the $k_{s}^{+}$ for both current alone flow and wave current combined flow. The influence of $k_{s}^{+}$ due to the mutual interplay between rib roughness and flow condition may modify the near wall velocity and turbulence characteristics.

Figure 6.

Mean velocity profiles in inner coordinates. Solid lines are log-law (U⁺ = 2.44lnY⁺+ 5) for smooth wall surface: (a) CA, (b) WCC_0.4, and (c) WCC_0.8.

Results and discussions

Mean velocity distribution

The streamwise mean velocity profiles in outer coordinates for all the test conditions are shown in Figure 7. The measured freestream velocity ( $U_{e}$ ) is used to normalize the mean velocity at each measured location. The normalized mean velocity ( $\bar{u} / \bar{u} U_{e} U_{e}$ ) are plotted against y/B as shown in Figure 7(a) to (c). The y/B is the normalized distance of measurement location along the transverse direction of channel from the side wall boundary to centerline of the channel. The closest measurement location of channel side wall is y/B = 0.015 and y/B = 1 represents the centerline location. Apparently, the measurement locations closest to the smooth side wall usually represent the measurement locations inside the cavity region when rib roughness attached to the smooth side wall. Hereafter in Figure 7 the data points at y/B = 0.015 and 0.03 represent the measurement locations inside the cavity region and y/B = 0.05 represent the rib-crest line for considered p/k = 2, 3, and 4 rib roughness in this experimental study. Thus, it is to be noted that the velocity profiles for smooth wall surface (SM_h2) and rough wall surfaces (R1_h2, R2_h2, and R3_h2) which is shown in Figure 7(a) to (c) were measured at same locations for both the smooth and rough wall cases. The same protocol was followed for all three flow conditions CA, WCC_0.4, and WCC_0.8.

Figure 7.

Normalized mean velocity profiles over smooth and rough surfaces for (a) CA, (b) WCC_0.4 and (c) WCC_0.8 flow cases in outer coordinates and velocity gradient profiles for (d) CA, (e) WCC_0.4, and (f) WCC_0.8.

Figure 7(a) shows that mean velocity profiles over rough walls have smaller magnitudes than smooth wall profiles toward the wall boundary, indicating that roughness has a significant impact on the shape of velocity profiles near the wall boundary. The velocity decreases due to momentum deficit caused by the roughness.⁶¹ Similar results are found for wave-current combined flow (Figure 7(b) and (c)). Figure 7(a) to (c) reveals that the major deviation of mean velocity profiles for rough walls compared to that for smooth wall profiles are nearly upto y/B = 0.3 and beyond y/B>0.3 the same deviation is almost negligible. As a result, the velocity profile for smooth wall and rough walls are almost same in the outer region of the boundary layer. These observations imply that the impact of p/k = 2, 3, and 4 roughness is limited to the region of y/B<0.3 for the presented current alone and wave-current combined flow. The roughness sublayer is defined as the region typically 5 times of roughness height from the wall in the rough-wall turbulent boundary layer¹¹ and in the present experiment this roughness sublayer is found based on above as y/B<0.25 region of the boundary layer. Thus it is evident from Figure 7(a) to (c) that the effect of p/k = 2, 3, and 4 (d-type) rib roughness on the mean velocity profiles are majorly confined to the roughness sublayer and beyond this sublayer the profiles are almost collapsed. Li and Li⁶⁵ reported in their study that within the roughness sublayer of d-type roughness, the velocity profile follows a quasi-linear distribution and beyond the roughness sublayer multiple profiles collapse into one. It is clear from Figure 7(a) to (c) that the negative velocities occur inside the ribs’ cavity region representing the flow reversal.⁶⁶ The magnitude of maximum negative velocity for current alone flow is about 3% of the free stream velocity whereas for wave-current combined flow it is about 7% of free stream velocity. The magnitude of negative velocity is relatively high for wave-current combined flow compared to current alone flow and may be due to the presence of orbital velocity in the wave-current combined flow condition.^3,24

The mean velocity gradient, $d \bar{u} / d \bar{u} dy dy$ are computed using second-order central difference method¹⁴ for smooth wall and rough wall cases. The values of $d \bar{u} / d \bar{u} dy dy$ for smooth (SM) and rough (R1, R2, and R3) surfaces are plotted against the y/B in Figure 7(d) to (f) for CA, WCC_0.4 and WCC_0.8 respectively. Characterization of the gradient is important for understanding the mean momentum transportation as well as the production of turbulent kinetic energy and Reynolds stresses.¹⁴ Figure 7(d) to (f) reveal that the values of gradient for smooth wall case is fairly constant across the boundary layer in region y/B>0.1 and for rough wall cases that is found at y/B>0.2 region for both current alone flow and wave current combined flow. Also it may be noted that beyond y/B≈0.2 both the smooth and rough wall profiles almost collapsed over each other. Thus it may be concluded that the roughness effect in the present experiment for both current alone flow and wave current combined flow remain in the range of y/B≈0.2 of the boundary region which is less than the 5 times of rib roughness height (y/B = 0.25). For CA flow, the $d \bar{u} / d \bar{u} dy dy$ profile against y/B under the roughness effect gradually decreases and results in profile over rough wall that almost collapsed with the profile over smooth wall with increasing distances from the boundary (Figure 7(d)). For WCC_0.4 and WCC_0.8 flow cases the $d \bar{u} / d \bar{u} dy dy$ profile against y/B shows similar results under the effect of roughness as observed for CA flow with relatively higher gradient values.

Turbulence intensities

Figure 8 shows the profiles of normalized turbulence intensity in the streamwise direction, $I_{u}^{+}$ and wall normal direction, $I_{v}^{+}$ over smooth (SM) and rough walls (R1, R2, and R3) for CA, WCC_0.4 and WCC_0.8 flow cases. Turbulence intensities are normalized by the shear velocity, $U_{τ}$ . The normalized turbulence intensities $I_{u}^{+}$ ( $= I_{u} / I_{u} U_{τ} U_{τ}$ ) and $I_{v}^{+}$ ( $= I_{v} / I_{v} U_{τ} U_{τ}$ ) are plotted against the y/B as shown in Figure 8. Noticeable modulation in the turbulence intensity profile is observed for CA, WCC_0.4 and WCC_0.8 flow cases within the y/B<0.25 region. The Figure 8(a) shows the intensity profiles for smooth surface and Figure 8(b) to (d) show the same for rough (R1, R2, and R3) surfaces. The comparisons of profiles between smooth (Figure 8(a)) and rough surfaces (Figure 8(b)–(d)) show the distinct features of profiles. In smooth wall surface (Figure 8(a)) the $I_{u}^{+}$ profiles along the y/B for wave current combined flows (WCC_0.4 and WCC_0.8) are weaker than the current alone flow (CA) specifically near the boundary wall. In a similar way, the streamwise turbulence intensity weakened when wave was superimposed to the current flow was reported in the experimental study of turbulence structure under wave-current motion by Umeyama.⁶⁷ Figure 8(a) shows that the largest value of $I_{u}^{+}$ occurs adjacent to the wall. The values of $I_{u}^{+}$ for WCC_0.8 are relatively lesser than WCC_0.4 at all measured y/B locations for smooth wall (Figure 8(a)). In connection with the above results Kemp and Simons³ and Umeyama⁶⁷ documented similar features for bottom boundary layer and reported that larger wave heights gave smaller values of turbulence intensity. In the present experimental conditions, the addition of wave frequency 0.8 Hz in the current alone flow (i.e. WCC_0.8 case) attributed larger wave height (Table 1) than the addition of 0.4 Hz wave frequency. For the wall with d-type rib roughness the peak value of streamwise turbulence intensity ( $I_{u}^{+}$ ) profiles is observed at the proximity of the plane of rib top for all the rough wall (R1, R2, and R3) cases for both current alone flow and wave-current combined flow (Figure 8(b)–(d)). This is probably due to the formation of shear layer as flow separates from the leading edge of the rib.⁶¹ The d-type rough bed flows are characterized as the flows with isolated stable recirculation vortices in the cavities between the roughness elements.^68,69 Figure 8(b) to (d) reveal that the peak value increases with the increase of spacing between two ribs within the d-type ( $p / p k < 5 k < 5$ ) rib-roughness for CA flow condition. Aghaei Jouybari et al.⁷⁰ studied the turbulence structure over cube roughness with p/k = 3 in open channel flow and the results suggest that significant amounts of vortices at crest location of roughness and have substantial influence on turbulence intensities and shear production.

Figure 8.

Normalized streamwise turbulence intensity profiles over (a) smooth and (b-d) rough surfaces for CA, WCC0.4 and WCC0.8 flow cases. And normalized wall normal turbulence intensity profiles over (e) smooth and (f-h) rough surfaces for CA, WCC0.4 and WCC0.8 flow cases.

For wave-current combined flow (both WCC_0.4 and WCC_0.8) over the d-type rough side wall (Figure 8(b)–(d)) the streamwise turbulence intensity $I_{u}^{+}$ profile shows higher values than the $I_{u}^{+}$ profile of current alone flow in the roughness sublayer. Therefore, it may be conceived that the turbulence intensities are more affected by the side wall roughness when wave is superimposed on current alone flow. Dingemans et al.⁷¹ reported two counter rotating vortices in the cross section of channel due to the sidewall effect when waves are added to the current flow. Moreover Klopman⁷² provided the evidence of such vortices in the wave current combined flow. The flow turbulence strongly depends on the interactions between the unsteady vortices shed from roughness elements and the flow above.⁵ Waves also enhance the turbulence generation at the bed by increasing the effective bottom roughness felt by the current.²⁰ As vortices are shed and interact with roughness elements during the wave cycle, the turbulence properties change significantly.²⁴

In the rough boundary wall, the addition of even smallest wave height with the current alone flow caused a spectacular increase in turbulence intensity compared to that of current alone flow.³ The addition of waves to the current probably imposes additional turbulence scales through nonlinear interaction with the turbulence field of the current alone flow under roughness influence resulting in enhanced turbulence intensity at the near side wall region. Pertinent to these Figure 8(b) to (d) shows that large $I_{u}^{+}$ in the roughness sublayer for wave-current combined flow compared to current alone flow. Qualitatively it highlights that the increasing rate of near wall peak value of streamwise turbulence intensity ( $I_{u}^{+}$ ) for interaction between rib roughness and wave current combined flow is relatively higher than the interaction between rib roughness and current alone flow. The cavity region of d-type rib roughness occupied by the circulation region⁶⁹ and probably this circulation region interacts with the orbital velocity of wave current combined flow²⁴ that may enhance the turbulence intensity. Moreover Figure 8(b) to (d) show larger $I_{u}^{+}$ values within the roughness sublayer for R3 compared to R1 and R2 which indicates that $I_{u}^{+}$ increases with the increase in the spacing between the rib elements for d-type rib roughness. As discussed earlier, with the increase of p/k ratio of the rib wall roughness in the wave current combined an increase in the roughness scale is pertinent (i.e. equivalent sand grain roughness.¹

The wall normal turbulence intensity ( $I_{v}^{+}$ ) profiles for CA flow overall rough boundary cases studied herein (Figure 8(f)–(h)) shows highest magnitude of $I_{v}^{+}$ near the wall and it gradually decreases up to the roughness sublayer (y/B<0.25). Beyond the roughness sublayer $I_{v}^{+}$ values are almost of constant magnitude. For smooth wall surface boundary and CA flow, the $I_{v}^{+}$ profiles are relatively of constant magnitude throughout the boundary layer (Figure 8(e)). The $I_{v}^{+}$ profiles in the case of wave-current combined flow (WCC_0.4 and WCC_0.8) over smooth and R1, R2, and R3 rough surfaces show similar pattern as the profiles of CA for smooth and rough surfaces (Figure 8(e)–(h)). In Figure 8(b) to (d) and (f) to (h) it is reveal that the highest magnitudes of $I_{v}^{+}$ within the roughness sublayer for R1, R2 and R3 rough surfaces lie in the range 22%–31% of $I_{u}^{+}$ for CA flow case, 23%–30% of $I_{u}^{+}$ for WCC_0.4 and 22%–27% of $I_{u}^{+}$ for WCC_0.8 flow cases. These range of values of $I_{v}^{+}$ in the near wall region of ribbed rough surfaces for CA, WCC_0.4 and WCC_0.8 flow cases are quite significant for the streamwise turbulent flow. Probably the near wall turbulent fluctuations in the wall normal direction are influenced by the rib roughness elements which resulted in the increase in wall normal turbulence intensity significantly.

Reynolds shear stress

In Figure 9 the profiles of normalized Reynolds stress ( $R_{uv}^{+}$ ) are shown for current alone flow and wave-current combined flow. Reynolds shear stress is normalized by the shear velocity ( $R_{uv}^{+} = R_{uv} / R_{uv} U_{τ}^{2} U_{τ}^{2}$ ) and plotted against y/B. Figure 9(a) show the distribution of $R_{uv}^{+}$ over smooth side wall and Figure 9(b) to (d) show over rib roughness side wall. The $R_{uv}^{+}$ profiles were measured for rough boundary wall conditions from the rib cavity center to channel center line along transverse direction. For smooth wall surface, the $R_{uv}^{+}$ profiles were measured at the same locations as that measured for rough boundary wall. These results are compared with the results of rough wall surface for CA, WCC_0.4, and WCC_0.8 flow cases. Figure 9(a) shows that the magnitude of the Reynolds shear stress gradually increases and reach a peak value in the near wall region and thereafter it gradually decreases almost upto the y/B = 0.3 and then took nearly constant values in the outer layer for smooth wall boundary. For rough walls cases (Figure 9(b)–(d)) the peak value is obtained close to the top of the rib surface and gradually decreases in the outer layer, reflecting the influence of wall roughness. Close inspection of the shear stress profiles of Figure 9(b) to (d) reveals that the peak value of shear stress lies almost in rib crest line and below the rib crest line the shear stress value decreases toward the wall. Li and Li⁶⁵ reported that the highest spatial average value of turbulence shear stress occurs at the crest level of roughness elements. The Figure 9(b) to (d) gives a clear indication of shearing flow caused by the velocity difference³⁶ in the rib crest line because below the rib crest line that is, cavity region is occupied by flow reversal with small magnitude of velocity.⁶⁹ The magnitudes of $R_{uv}^{+}$ is relatively large for rough walls cases compared to smooth wall case. Moreover, the magnitudes of $R_{uv}^{+}$ within the roughness sublayer for R2 and R3 roughness cases showed comparatively large values relative to that for the R1 roughness case indicating the influence of roughness on Reynolds stress. The pattern of the distribution of $R_{uv}^{+}$ for both current alone flow and wave current combined flow shows similar nature. For the smooth wall case (Figure 9(a)) $R_{uv}^{+}$ values are observe to be lesser for wave current combined flow than the current alone flow. This result agrees well with the distribution of Reynolds shear stress for the smooth boundary as reported in Umeyama⁶⁷ and Kemp and Simons.³ For the rough wall cases (Figure 9(b)–(d)) the magnitudes of $R_{uv}^{+}$ at near wall region for wave-current combined flows are larger than the current alone flow. These results indicate that the Reynolds shear stresses are strongly influenced by the mutual interaction of wave-current flow and rib roughness elements. The vortices of wave-dominated flow interact with the vortices formed by the interplay between roughness elements and wave-current dominated flow. Also from Figure 9(b) to (d) it is reveal that the magnitudes of $R_{uv}^{+}$ at near wall region increases with the increase of rib spacing for both current alone flow and wave-current combined flow for the presented d-type rib roughness.

Figure 9.

Normalized Reynolds shear stress profiles over smooth and rough surfaces for CA, WCC_0.4 and WCC_0.8 flow cases: (a) SM_h2, (b) R1_h2, (c) R2_h2 and (d) R3_h2.

TKE dissipation rate (ε) estimation

In order to understand the effect of d-type rib roughness under the wave-current combined flow on TKE (turbulent kinetic energy) dissipation rate (ε), equation (15) is used to estimate the TKE dissipation rate assuming the local isotropy.⁷³

ε = \frac{15 ν}{U^{2}} \bar{(\partial u / \partial u \partial t)^{2} \partial t)^{2}}

(15)

Thereafter, the obtained TKE dissipation rate, ε, is made dimensionless by B and U_τ, $ε^{+} = ε B / ε B U_{τ}^{3} U_{τ}^{3}$ . Figure 10 shows the profiles of $ε^{+}$ as function of wall normal distances, y/B over smooth wall and R1, R2, and R3 rough wall cases for current only flow, CA and wave-current combined flow, WCC_0.4 and WCC_0.8.

Figure 10.

Profiles of normalized TKE dissipation rate, ε+ = εB/U3 τ over smooth and rough surfaces for (a) CA, (b) WCC_0.4, and (c) WCC_0.8 flow cases.

Figure 10 reveals that for rough surfaces R1, R2, and R3, the magnitudes of $ε^{+}$ near the wall boundary are significantly greater in comparison to smooth wall boundary. In the outer boundary of the wall, no clear distinct features are observe and the $ε^{+}$ profiles of smooth and rough surfaces for CA, WCC_0.4 and WCC_0.8 flow cases almost collapses with each other. For CA flow case, the magnitudes of $ε^{+}$ for R1, R2, and R3 are greatest near the wall and gradually decrease and from about y/B≈ 0.1 it follows almost similar profile of that for the smooth wall. In general, the dissipation rates are typically low in the outer boundary layer and significantly higher in the roughness crest layer due to the turbulence caused by the roughness wakes.³⁵ For WCC_0.4 and WCC_0.8 flow cases, beyond y/B≈ 0.2, the magnitudes of $ε^{+}$ for R1, R2, and R3 collapse with smooth wall profile (Figure 10(b) and (c)). Greater magnitudes of $ε^{+}$ for R1, R2, and R3 near the boundary region reveals that the TKE dissipation rate increases systematically due to d-type (p/k = 2, 3 and 4) rib roughness compared to smooth side wall surface. Moreover by critically evaluating the region y/B = 0 to 0.25 in Figure 10(a) to (c), it is reveal that the roughness effects on $ε^{+}$ profiles are limited within the roughness sublayer (y/B<0.25) for CA, WCC_0.4 and WCC_0.8 and beyond the roughness sublayer (y/B>0.25) the profiles for smooth and rough surfaces are almost same. In Figure 10(a) the distribution of the magnitudes of $ε^{+}$ for R1, R2, and R3 at near wall region is in the order R1 < R2 < R3 which implies that TKE dissipation rate increases with the increase of pitch-to-height ratio (p/k) for d-type roughness. This result agrees with the result reported in Agelinchaab and Tachie⁶⁶ over smooth surface and p/k = 2, 4, and 8 type rough surfaces. For wave-current combined flow for WCC_0.4 and WCC_0.8 cases (Figure 10(b) and (c)) the profiles of distribution of $ε^{+}$ , smooth and rough surfaces show similar trend as that for CA flow case. The magnitudes of $ε^{+}$ are slightly greater for WCC_0.4 (Figure 10(b)) and WCC_0.8 (Figure 10(c)) flow cases at most of the measurement locations across the boundary layer compared to that for CA (Figure 10(a)) flow case. Probably the mutual interplay between rib roughness and wave current combined flow results in increase rate of dissipation of energy for wave current combined flow. Singh et al.²⁶ reported in their study that dissipation rate slightly increases for wave-current combined flow compared to current alone flow. From the Table 2, it reveals that the typical roughness scale equivalent sand-grain roughness height in the rough wall bounded turbulent flow increases with the interaction of p/k roughness and wave-current flow compared to that of the interaction of p/k roughness and CA flow. Moreover with the increase of wave frequency for wave current combined flow, the equivalent sand-grain roughness height increases in the wall boundary layer in the wave dominated flow. This result may be the reason for the occurrence of comparatively higher dissipation rate for WCC_0.8 than WCC_0.4 at near wall region. Yu et al.³⁵ reported that dissipation rates over rough boundary is higher for larger wave. In the present experiment, WCC_0.8 attributes wave current combined flow with larger wave height. It may be noted here that the literatures^26,66,73 suggests that the distribution of dissipation rate is affected by the surface roughness and slightly increases with the increase of surface roughness.

Turbulent length scales

In view of the energy cascade introduced by Richardson,⁷⁴ the turbulence can be considered to be composed of eddies of different sizes. The various sizes of eddies are characterized by the different scales of motion. The largest size of eddies are defined by the integral length scale at which the kinetic energy is produced. According to Richardson,⁷⁴ the large eddies are unstable and break up into smaller eddies by transferring their energy. The smallest size of eddies are defined by the Kolmogorov length scale and at this length scale the energy is dissipated. The intermediate size between largest and smallest size eddies are known as Taylor scale.⁷⁵ The physical interpretation of Taylor scales is not so clear. Mainly it is used to define the Taylor scale-Reynolds number which is used in grid generated turbulence.⁷⁵ Kolmogorov, Taylor and integral length scales play an important role to characterize the structure of turbulence. The present study focuses on the distribution of typical turbulent scales across the side wall boundary layer under both current alone flow and wave-current combined flow over smooth and d-type rough boundary. The present study also emphasizes on the effect of increasing spacing between two ribs in the side wall rib roughness and a comparison is made with the without rib-roughness side wall in the context of changes observed in the turbulence length scales.

The streamwise integral length scale (L_u) is determined following the estimation of the integral time scale. The integral time scale is calculated by integrating the data $τ = 0$ to the first zero crossing point of the autocorrelation function and calculating the area under the autocorrelation function. For example, Figure 11 represents the graph of autocorrelation function ( $R_{uu}$ ) versus lag ( $τ$ ) of test run SM_h2 at y/B = 0.03 for CA flow case and in the figure the circle denotes the first zero-crossing point of $R_{uu}$ . The autocorrelation of a random process describes the correlation between values of the process at distinct points in time as a function of the two times or of the time gap. It depicts the correlation between the consecutive values of the time series.

Figure 11.

Streamwise autocorrelation function for test run SM_h2 at y/B = 0.03 for CA flow case. Dashed line represents the zero-line. Circle denote the first zero-crossing point of Ruu.

The streamwise velocity component $u (t)$ is employed to estimate the autocorrelation function ( $R_{uu}$ ) and integral time scale ( $T_{u}$ ). The mathematical expressions of $R_{uu}$ and $T_{u}$ are

R_{uu} (τ) = \frac{u' (t) u' (t + τ)}{u' (t) u' (t)}

(16)

T_{u} = \int_{τ = 0}^{τ = (R_{uu} (τ) = 0)} R_{uu} (τ) d τ

(17)

where $u'$ represents the turbulent fluctuation part and $τ (= 0.025 s)$ is the time lag. Using Taylor’s frozen turbulence hypothesis, the stream wise integral length scale ( $L_{u}$ ) corresponds to the size of fluctuating eddy motions that exist in turbulent flows and is obtained as $L_{u} = T_{u} \bar{u}$ where $\bar{u}$ is the streamwise mean velocity. The L_u is normalized by B such as $L_{u}^{+} = L_{u} / L_{u} B B$ and plotted against the normalized wall normal distances y/B.

Figure 12 shows the distribution of $L_{u}^{+}$ over smooth and rough surfaces for CA, WCC_0.4 and WCC_0.8 flow cases. For CA flow case (Figure 12(a)) $L_{u}^{+}$ values over the rough surfaces at near wall region (y/B<0.3) are considerably smaller than that over the smooth surface. But far away from the near wall region, the distribution of $L_{u}^{+}$ are almost identical for both smooth and rough surfaces indicating the strong effect of rib roughness on streamwise integral length scales at the near wall region. Apparently in the near wall region, the wall roughness cases (p/k = 2, 3, and 4) with rib play a dominant role on the size of turbulent eddies. Moreover Figure 12(a) shows that $L_{u}^{+}$ decreases with the increase of the spacing between two ribs and these changes are limited to the roughness sublayer. However, the magnitude of $L_{u}^{+}$ in the cavity region exhibits be to independent of the spacing between ribs for R1, R2, and R3 roughness for CA flow condition as well as when waves (f = 0.4 and 0.8 Hz) added to the CA flow. Interestingly, while wave frequency 0.4 and 0.8 Hz are added to the current flow individually, the trend of distribution of $L_{u}^{+}$ deviates for each case from the trend obtained for CA flow. Figure 12(b) and (c) show the profiles of $L_{u}^{+}$ for wave –current combined flow conditions WCC_0.4 and WCC_0.8 respectively. For both the conditions, the magnitudes of $L_{u}^{+}$ in the profiles over the boundary layer gradually increase and reach a peak value and thereafter gradually decrease moving toward the center line of channel from the wall. The peak value as well as other measured data points of the profiles of $L_{u}^{+}$ is smaller for WCC_0.8 than that for WCC_0.4. Furthermore, the values of $L_{u}^{+}$ for both the WCC_0.4 and WCC_0.8 cases are smaller than the CA flow case. Thus the magnitude of the modulation of the streamwise integral length scale profiles increases with the increase in wave frequency of the wave current combined flow. This result agrees well with Petti and Longo⁷⁶ in connection to the distribution of integral length scales for different wave frequencies. Figure 12(a) to (c) show that a large values of $L_{u}^{+}$ over smooth wall for CA, WCC_0.4, and WCC_0.8 flow condition is observed at ∼y/B = 0.2 for WCC_0.4 and WCC_0.8 cases. The high value at ∼y/B = 0.2 can be attributed to the effect of the anisotropy.⁷⁶

Figure 12.

Normalized streamwise integral length scale profiles over smooth and rough surfaces for (a) CA, (b) WCC_0.4 (c) WCC_0.8 flow conditions.

The smallest size of eddies known as Kolmogorov length scale (η) is calculated from the estimates of dissipation rate (ε) obtained from equation (15). The Kolmogorov length scale is given by $η = {(ν^{3} / ν^{3} ε ε)}^{1 / 4}$ where, $ν$ is kinematic viscosity⁷⁷ and the normalized Kolmogorov length scale is given by $η^{+} = η / η B B$ . The profiles of $η^{+}$ are plotted against the y/B for CA, WCC_0.4, and WCC_0.8 flow conditions and are shown in Figure 13(a), (b) and (c) respectively. All the profiles were taken over the smooth surface and the rough surfaces R1, R2, and R3. Figure 13(a) shows the profiles of $η^{+}$ over smooth and rough surfaces for CA flow case. Similarly, Figure 13(b) and (c) show the profiles of $η^{+}$ for WCC_0.4 and WCC_0.8 flow cases respectively. The trend of profiles of $η^{+}$ are reasonably similar to the trend of $L_{u}^{+}$ for similar background conditions as documented in Figure 12 and it is expected to follow the following relation as reported in Tennekes and Lumley.⁷⁷

\frac{η}{L_{u}} = S^{- 1 / - 14 4} {Re}_{L_{u}}^{- 3 / - 34 4}

(18)

where $S \approx 1$ and ${Re}_{L_{u}} = (\hat{u} L_{u}) / (\hat{u} L_{u}) ν ν$ , ( $\hat{u}$ is velocity scale, $L_{u}$ is length scale and $ν$ is kinematic viscosity). The magnitudes of $η^{+}$ in the profiles gradually decrease as it approaches the side wall boundary. The $η^{+}$ is quite different near the smooth and rough side wall boundary. Near the side wall boundary of R1, R2, and R3 rough surfaces, the $η^{+}$ is lower than the smooth surface wall and away from the wall the $η^{+}$ is quite similar to the smooth wall for the current alone flow case (Figure 13(a)). But for the wave current combined flow cases (Figure 13(b) and (c)) the $η^{+}$ profiles are distinctly separated from the current alone flow at far away region of the wall. The ranges of the values of $η^{+}$ in the profiles are a little smaller over the R1, R2, and R3 surfaces compared to that over the smooth surface. These results indicate that a reduction in energy of the dissipative scales occur near the side wall region with the introduction of rib roughness in the smooth side wall boundary. Further reduction in these scales occurs in the wave current combined flow under the mutual interplay between wall roughness and wave induced flow.

Figure 13.

Normalized Kolmogorov length scale profiles over smooth and rough surfaces for (a) CA, (b) WCC_0.4 (c) WCC_0.8 flow conditions.

Figure 14 shows the profiles of normalized Taylor length scale ( $Λ^{+}$ ) which represents the intermediate size of eddies between largest (integral length scale) and smallest (Kolmogorov length scale) size of eddies. The Taylor scales are estimated using the Kolmogorov’s second similarity hypothesis⁷⁵ given by

Λ = \sqrt{\frac{15 ν σ_{u}^{2}}{ε}}

(19)

Here, $σ_{u}$ is the standard deviation in of velocity fluctuation in the streamwise direction. Thereby $Λ^{+} = Λ / Λ B B$ is plotted as a function of y/B and are shown in Figure 14(a) to (c) for CA, WCC_0.4, and WCC_0.8 flow conditions respectively. The profiles were measured over smooth surface and R1, R2, and R3 rough surfaces for CA, WCC_0.4, and WCC_0.8 flow conditions. The trend of profiles of $Λ^{+}$ for CA, WCC_0.4, and WCC_0.8 are quite similar to the trend of $L_{u}^{+}$ and $η^{+}$ . The shape of profiles obtained for CA flow is not affected when waves are added to the CA flow but the range of magnitude of $Λ^{+}$ is reduced. This reduction in magnitude is more pronounced for WCC_0.8 than WCC_0.4, that is, the addition of relatively higher wave frequency of 0.8 Hz to the current flow reduces $Λ^{+}$ more significantly. The reduction of $Λ^{+}$ , may be due to the reduction of $σ_{u}$ (equation (19)) and it is already observed that $σ_{u}$ reduces with the increase of wave frequency (considered for the present study) in the wave current combined flow.

Figure 14.

Normalized Taylor length scale profiles over smooth and rough surfaces for (a) CA, (b) WCC_0.4 (c) WCC_0.8 flow conditions.

Streamwise integral, Kolmogorov and Taylor length scale distribution with respect to the wall normal distances (y/B) from the side wall of the channel are shown in Figures 12 to 14 respectively. The rate of transfer of turbulent kinetic energy and the energy dissipation rate can be related to the characteristic length scale for turbulent motion. It is observed that length scales in the proximity of side wall region for all roughness cases (p/k = 2, 3, and 4) are smaller than smooth surface that is, surface without rib roughness, which implies that rib roughness enhances the energy dissipation rate. Furthermore, with the increase of spacing between two ribs a decrease in the length scales is observed. This corresponds to the lower momentum transfer with the increase of energy dissipation rate, for the range of spacing considered in the present study.

Anisotropy of Reynolds stresses

It is of substantial interest to have some prior knowledge on the anisotropy of the flow field for the simulation of turbulent flow over rough surfaces computationally.⁷ This knowledge of anisotropy should be useful in identifying the sensitivity of turbulence structure for various boundary conditions, in addition to being relevant in the context of developing turbulence models.⁸ Also, the invariant analysis primarily provides information on the anisotropy of the large-scale motion^7,8 showed that the turbulence in a rough wall boundary layer is more isotropic than that in a smooth wall boundary layer.

To quantify the level of anisotropy of the turbulence field, Lumley⁷⁸ introduced the Reynolds stress anisotropy tensor ( $a_{ij}$ ) and its three invariants I, II, and III. The anisotropy tensor is obtained as follows.⁶

a_{ij} = \frac{\bar{u_{i} u_{j}}}{2 k} - \frac{δ_{ij}}{3},

(20)

and invariants

I = a_{ii}, II = a_{ij} a_{ji}, III = a_{ij} a_{jk} a_{ki}

(21)

Here $k = \frac{1}{2} \bar{{u'}_{i} {u'}_{i}}$ is the turbulent kinetic energy and $δ_{ij}$ is the Kronecker delta function ( $δ_{ij} = 1$ when $i = j$ and =0 when $i \neq j$ ). The anisotropy invariants map (AIM) is the cross plot of second ( $II$ ) versus third ( $III$ ) invariants of Reynolds stress anisotropy tensor, where $II$ represents the degree of anisotropy, while $III$ represents the nature of the anisotropy.⁷⁹ The AIM provides a convenient method to depict graphically any turbulence model’s performance in terms of the limiting states of the map and to serve as a tool to guide the development of turbulence models analytically.⁸⁰ The boundaries of the map as shown in Figure 15 represent the characteristic states of turbulence. There are two curves extending from the origin. Right curve corresponds to axisymmetric expansion in which one diagonal component of Reynolds stress tensor is larger than the other two equal components. The left curve represents an axisymmetric contraction, in which one component is smaller than the other two equal components. The limiting points of these curves are important. The origin that is, bottom cusp (II = 0, III = 0) of the map (Figure 15) represents the isotropic turbulence, the end of right curve and left curve represents the one-component and two-component turbulence respectively. Upper boundary line of the AIM (Figure 15) represents the two-component turbulence which is prevalent near the solid walls where the wall-normal component of the fluctuation vanishes much faster than the other components.^81,82

Figure 15.

Anisotropy invariant map (AIM).

In the present study, the AIM is presented to characterize the state of turbulence for smooth wall boundary and rough wall boundary (p/k = 2, 3, and 4) for both current alone (CA) flow and wave current combined (WCC_0.4 and WCC_0.8) flow. Figure 16 shows the data for CA flow case and Figure 17 shows the data for wave current combined (WCC_0.4 and WCC_0.8) flow cases. Data in Figures 16 and 17 represent different y/B locations considering transverse direction outward from the side wall to center line of channel for both smooth wall and rough wall boundary cases. Pertaining to the data points in Figures 16 and 17, location y/B = 0.015 represents the near wall data for smooth wall boundary and for rough wall cases it represents the data point that lies in the cavity region for p/k roughness within the roughness elements. Similarly the location y/B = 0.05 and y/B = 1 indicate the data point of rib-crest line and center-line of channel width respectively. Further y/B = 0.1 indicates the 1k (one times of rib height) distance from the rib-crest line which is within roughness sublayer. In the Figure 16, the magnitudes of –II and III differ considerably between smooth and rough surfaces for CA flow case. For the smooth surface, the tendency of data of AIM (–II and III) is to decrease with the increase in y/B moving outward from the wall. For the rough surfaces (R1, R2, and R3), the pattern of data of AIM is quite similar to that of smooth surface. Figure 16 shows smaller magnitudes of –II and III in the far wall region, which results in data concentration at the bottom cusp region of AIM. The data for location y/B = 1 is close to the axisymmetric line. This result agreed with study of Krogstad and Torbergsen⁷ invariant analysis of wall bounded turbulent flow by Krogstad and Torbergsen.⁷ They documented that the central region of a fully developed pipe flow follows the axisymmetric line and anisotropy increases toward the wall. Figure 16 reveals that close to the side wall, the data shift away from the axisymmetric line and approach two component turbulence showing increased magnitudes of –II and III. In this regard, the smooth wall has maximum magnitudes of –II and III at near wall region (y/B = 0.015) which corresponds to higher anisotropy compared to rough wall cases (R1, R2, and R3) as these rough walls experiences low magnitudes of –II and III. At y/B = 0.015, the state of turbulence largely vary for smooth surface and rib roughness because for rough surface this position corresponds to the region within cavity formed by rib roughness elements. Thus, it is evident from Figure 16 that the data within the cavity region follows the left axisymmetric turbulence line of the AIM closely and move toward the bottom cusp with the increase of y/B from 0.015 to 1. From Figure 16 it reveals that the shape of Reynolds stress changes to “sphere like” from “ellipse like” for smooth wall and from “disk like” for rough wall with the increase of distance from the side wall.⁷⁵ From Figure 16 it is further reveal that change in rib spacing shows low sensitivity to the change of anisotropic structure of turbulence. Further work probably is required with a greater number of variations of p/k ratio to understand the effect of rib roughness in modulation of the anisotropic structure of turbulence.

Figure 16.

Anisotropy invariant map (AIM) of Reynolds stress tensor for smooth and rough surfaces boundary wall for current alone (CA) flow.

Figure 17.

Anisotropy invariant map (AIM) of Reynolds stress tensor for smooth and rough surfaces boundary wall for wave current combined flow (a) WCC_0.4 and (b) WCC_0.8.

Figure 17(a) and (b) show the AIM for wave-current combined flow cases WCC_0.4 and WCC_0.8 respectively. For wave current combined flow cases, comparable features are observed as that for current alone flow case. Figure 17(a) and (b) show that majorly the data points are distributed in a wider range along the right axisymmetric line of the AIM compared to CA flow case (Figure 16) where mostly the data points are confined in a small region of AIM. This result (Figure 17) implies that when the wave frequency 0.4 Hz (WCC_0.4) and 0.8 Hz (WCC_0.8) are added individually to the current alone flow, the state of turbulence is modulated for each case showing larger magnitude of –II and III in AIM for wave current combined flow cases compared to that for CA flow case. Furthermore, the WCC_0.8 shows the higher magnitudes of –II and III compared to WCC_0.4 which implies that increased level of anisotropy dominates when higher wave frequency 0.8 Hz is superimposed to the current alone flow. Similar to CA flow case, for the wave current combined flow cases (WCC_0.4 and WCC_0.8) also in general, the magnitudes of –II and III data for smooth wall surface are significantly larger than that over rough wall surfaces. The data adjacent to the side wall (i.e. y/B = 0.015) starts from the threshold region of two component turbulence and move toward the right axisymmetric limit line following a certain trajectory with increasing distance from the side wall. It is evident from Figure 17(a) that at y/B = 1, the data follow the right axisymmetric line indicating the one component turbulence with “cigar-shaped” turbulence. In contrast with the smooth wall surface, the data adjacent to the side wall starts from the region close to the left axisymmetric limit line for ribbed wall cases. In the cavity region (y/B = 0.015 for ribbed rough surfaces) often the streamwise component of fluctuating velocity is smaller than the other two lateral components⁷ and this possibly leads to the data for this location to align with the left axisymmetric turbulence line of AIM. Moreover Smalley et al.⁸³ reported that the state of turbulence below the crest plane that is, cavity of d-type roughness approach axisymmetric turbulence. Further they also reported that at the near wall recirculation region, two components axisymmetric turbulence is approached. Apparently for WCC_0.8 flow cases (Figure 17(b)) the data follows similar trend as that for WCC_0.4 (Figure 17(a)) with the little modulation of the magnitudes of –II and III indicating increased anisotropy with the increase of wave frequency. Close inspection of Figures 16 and 17 reveal that the level of anisotropy of turbulence in streamwise direction over both the smooth wall surface and rough surfaces for wave current combined flow is more compared to current alone flow. This result agreed well with study of Paul et al.⁸⁴ The state of turbulence adjacent to the rib crest line (y/B = 0.05) for both the current alone and wave-current combined flow cases demonstrate a remarkable shift from left axisymmetric turbulence to right axisymmetric turbulence. This change of state of turbulence can be characterized as transformation from “pancake-shaped” turbulence to “cigar-shaped” turbulence according to Choi and Lumley’s⁸⁵ description of turbulence. The reason behind this transformation may be attributed to the flow separation from just above the roughness elements of d-type rib roughness in streamwise direction.⁶⁶

Conclusions

In this study, a background experiment for wave alone case is performed to compare the second-order Stoke wave theory apart from the actual experiment objective, that is, to understand the effect of d-type rib roughness side wall on the turbulence structure for wave current combined flow. The transverse profiles (i.e. proximity of side wall to half of channel width) of mean velocity, turbulence intensity and Reynolds shear stress have been presented for the wave-current combined flow over the side wall rib rough boundary in order to examine the turbulence characteristics due to combine effect of wave-current flow and rib spacing at near wall boundary. The rib spacing was varied such that pitch-to-roughness height (p/k) ratios were 2, 3, and 4 representing d-type roughness. Two different wave-current combined flow cases wave frequency 0.4 and 0.8 Hz were explored. The transverse profiles were explored for mid water depth over both smooth and rough (p/k = 2, 3, and 4) side wall surfaces and for both current alone flow and wave-current combined flows. The results for rough boundary wall and wave-current combined flow have been compared with the smooth boundary wall and current alone flow cases to illustrate the effect of rib spacing and wave-current combined flow. The study of Reynolds stress anisotropy and the turbulent length scale distributions over the smooth surface and p/k rib roughness surfaces for wave current combined flow enrich the existing knowledge on turbulence characteristics near side wall ribbed rough surface under wave-current combined flow. The major findings of the study are given below.

In the background experiment for wave alone case, the wave produced by the wavemaker (cylindrical plunger type) is verified with the second-order Stoke’s wave theory. The free surface elevation and longitudinal velocity obtained from experiment agreed well with analytical result of second order Stoke wave equation.

Equivalent sand-grain roughness height (k_s) increases with the p/k ratio for both current-only flow and wave-current combined flow. For a given p/k, wave-current combined flow yields higher k_s when compared with the current alone flow. Higher wave frequency (f = 0.8 Hz) leads to greater k_s (=1.6).

Presence of ribs reduces the streamwise mean velocity near the rough (ribbed) side wall of the channel as compared to a smooth wall case. This mean velocity deviation remains confined within a region (known as roughness sublayer) having approximate thickness 5k (where, k is roughness height). The mean velocity gradient, $d \bar{u} / dy$ , near the side wall increases in wave-current combined flow. The higher the wave frequency the greater will be the mean velocity gradient near the rough wall.

For a channel with smooth side wall, the streamwise turbulence intensity near the side wall is relatively lesser for wave current combined flow compared to current alone flow. For ribbed wall, the wave-current interacting flow induces higher turbulence intensities compared to that induced by current alone flow. Further pitch to roughness height ratio influences the streamwise turbulence intensity for both current alone flow and wave-current combined flow. Present experimental results reveal that the near wall turbulence intensity increases with the superposition of wave on current compared to current alone flow due to the mutual interaction between rib roughness and wave current combined turbulence field. This probably occurs due to the addition of turbulence scales due to the interaction of wave with the roughness elements on the background turbulence. Moreover, turbulence intensity increases sharply near the wall as p/k ratio increases, which are true for both current-only flow and wave-current combined flow. Variation of Rreynolds shear stress with p/k ratio and different wave-current combinations are similar to that of streamwise turbulence intensity. For rib roughness, the peak value of shear stress profile lies nearly along the rib crest line and on both side of the crest line the shear stress value decreases asymptotically.

Wall roughness leads to substantially high TKE dissipation rate in the near wall region, as shown in Figure 10. Since wave-current combination enhances the equivalent sand-grain roughness height (k_s), the dissipation rate also increases in both near wall (boundary layer) region as well as core region of the channel for wave-current interacting flows, particularly for rough side-wall cases.

The effects of side wall roughness and wave interaction on the streamwise integral length scale or the largest eddy size are demonstrated through Figure 12. It is evident that the smooth side wall results in highest integral length scale. As the roughness (p/k ratio) increases, integral length scale decreases for both current alone flow and wave current combined flow. One of the most interesting findings of the present study is that the addition of wave reduces the integral length scale for both smooth and rough wall cases. With an increased wave frequency (WCC_0.8 case) the largest eddy size is reduced even further. The estimates of smallest or Kolmogorov length scale demonstrate similar effects of wall roughness and wave.

The results of anisotropy invariant map (AIM) of Reynolds stress tensor shows that near wall anisotropy reduced when p/k roughness introduced to the smooth side wall. The data adjacent to the side wall is two component turbulence and moves to axisymmetric turbulence with the increase of distances from the wall which imply the change of state of turbulence.

The wave current combined flow produces more anisotropy which reflects in the AIM where the data are distributed in a wide range along the right axisymmetric curve of AIM rather confined in a small region near bottom cusp of this map like current alone flow.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the Science and Engineering Research Board, Department of Science and Technology, Government of India for the financial support for this research (Contract no: EMR/2015/000266).

ORCID iDs

Koustuv Debnath

Debashis Pal

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