This study focuses on incompressible viscous fluid flow. The governing equations are the dimensionless continuity Equation (1) and the Navier-Stokes Equation (2), with viscous dissipation neglected.

$\nabla \cdot u=0$ (1)

$\frac{\partial u}{\partial t}+\nabla \cdot \left(uu\right)=-\nabla p+\frac{1}{Re}{\nabla }^{2}u$ (2)

Here, dimensional quantities are represented using an asterisk (*), and the equations are non-dimensionalized as $t=t^{*}{U}^{*}/H^{*}$, $x=x^{*}/H^{*}$, $p=p^{*}/\left(\rho U^{*2}\right)$, $u=u^{*}/U^{*}$.

Figure 1 shows the channel geometry and coordinate system. The left diagram is the front view, and the right diagram is the side view. The channel features a rectangular groove on the central lower side of the parallel plates. Using the origin $\text{O}$ as a reference, a right-handed Cartesian coordinate system is defined. The $x$-axis lies along the centerline of the groove’s bottom, the $y$-axis is normal to the parallel plates intersecting the $x$-axis at the left end of the channel, and the $z$-axis is along the spanwise direction. The main flow is assumed to be along the $x$-axis.

The channel geometry is approximately the same as those used in the experiments [5] and three-dimensional simulations [7] . The channel’s total length is $L=4.8$, the channel height in the no grooved region is $H=1$, and the depth is $D=11.4$. The groove length and height are set to $l=3.2$ and $h=0.8$, respectively.

Periodic boundary conditions expressed by Equation (3) were imposed in the main flow direction, along with a constant flow rate condition:

$\begin{array}{l}u\left(x+L,y,z,t\right)=u\left(x,y,z,t\right),\\ p\left(x+L,y,z,t\right)=p\left(x,y,z,t\right)-\text{Δ}p.\end{array}$ (3)

Here, $\text{Δ}p$ represents the mean pressure drop. A fixed pressure of $p=0$ was set at the point $\left(x,y,z\right)=\left(4.8,1.3,0\right)$.

Boundary conditions at both ends in the spanwise direction included three types: 1) wall boundary conditions, 2) periodic boundary conditions, and 3) two-dimensional conditions. The periodic boundary conditions in the spanwise direction are defined by Equation (4):

$\begin{array}{l}u\left(x,y,z+D/2,t\right)=u\left(x,y,z-D/2,t\right),\\ p\left(x,y,z+D/2,t\right)=p\left(x,y,z-D/2,t\right).\end{array}$ (4)

For wall boundary conditions, the velocity $u$ was set to a no-slip condition, and the pressure $p$ was specified to have a zero normal gradient. The remaining boundary surfaces also employed these wall boundary conditions. In two-dimensional conditions, physical quantities are assumed to be invariant in the $z$-direction.

In this study, the open-source software OpenFOAM 3.0.1 was used to discretize the governing equations with the finite volume method and perform unsteady numerical simulations [15] [16] . The application of this software to three-dimensional flows in sudden expansion channels within the transitional regime has been previously reported [17] - [19] .

The numerical scheme employed a second-order Crank-Nicholson method for time integration and second-order central differences for advection, viscous, and pressure terms, ensuring overall second-order accuracy. The PIMPLE algorithm, which combines the PISO and SIMPLE methods, was used for pressure-velocity coupling. The solvers applied were the geometric-algebraic multi-grid (GAMG) method for pressure and the Gauss-Seidel method for velocity. Iterative calculations continued until the residuals of velocity divergence decreased to 10⁻¹⁰. Although OpenFOAM allows for the use of various iterative solvers, we have confirmed that while the choice of solver may affect computational time, it does not influence the simulation results themselves.

As shown in Figure 2 , a non-uniform mesh with dense spacing near walls was employed for discretization. Three types of meshes were prepared for cases with wall boundary conditions in the spanwise direction. The number of mesh elements was 31 × 22 × 108 for Model A, 42 × 30 × 108 for Model B, and 42 × 30 × 153 for Model C. Figure 2 illustrates Model C. These mesh configurations were determined based on previous calculations for steady flows [7] , with Model B being the closest to those used in prior studies. The non-uniform mesh had a maximum-to-minimum spacing ratio of 8, with spacing changing geometrically. For cases with periodic boundary conditions in the spanwise direction, the mesh was uniform in the spanwise direction, with a configuration equivalent to 42 × 30 × 137 for Model C. For two-dimensional conditions, the mesh configuration was 42 × 30, equivalent to Model C.

The time step was set to $\text{Δ}t=0.002$ to maintain a maximum Courant number of $C{o}_{\max }\approx 0.2$. Figure 9 compares the distribution of the time-averaged velocity component $\langle u\rangle$ at $Re=350$ and $Re=658$ using the three different mesh models. No qualitative differences were observed in the flow characteristics due to the variation in mesh resolution. Since the flow patterns did not vary significantly among the three mesh types, the results obtained with Model C are presented unless otherwise noted.

The flow patterns are classified into four types: two-dimensional steady flow (A), two-dimensional oscillatory flow (B), three-dimensional steady flow (C), and three-dimensional oscillatory flow (D). The three-dimensional steady flow is observed only under wall boundary conditions and is symmetric in the $xy$-plane at $z=0$. Three-dimensional oscillatory flows are further categorized into symmetric flows (D.1), where $w=0$ in any $xy$-plane, and asymmetric flows (D.2), where $w\ne 0$ in all $xy$-planes. The flow symmetry in any $xy$-plane is expressed by Equation (5):

$\begin{array}{l}u\left(+z\right)=u\left(-z\right),v\left(+z\right)=v\left(-z\right),\\ w\left(+z\right)=w\left(-z\right),p\left(+z\right)=p\left(-z\right).\end{array}$ (5)

Here, “any $xy$-plane” refers to the $xy$-plane at $z=0$ for wall boundary conditions, while it includes all $xy$-planes at any $z$ for periodic boundary conditions.

Figure 3 shows the distribution of the velocity component w along a line at $z$-axis for $\left(x,y\right)=\left(3.2,0.4\right)$. At a given moment, it is used to examine the symmetry in the $xy$-plane. The moment when $w$ is presumed to have the largest amplitude was selected visually. Below, the changes in symmetry with respect to spanwise boundary conditions and Reynolds numbers $Re$ are described.

Under wall boundary conditions, at $Re=350$, the flow is symmetric with $w=0$ at $z=0$, but the symmetry is lost at $Re\ge 389$. For periodic boundary conditions, at $Re=350$, the flow is two-dimensional with $w=0$ at all $z$. At $Re=389$, the flow is three-dimensional symmetric with $w=0$ at $z=0$. For $Re\ge 428$, the flow becomes asymmetric with $w\ne 0$ in all $xy$-planes. The symmetric center at $z=0$ for $Re=389$ under periodic boundary conditions resulted from computational constraints, as the computational domain is symmetric at $z=0$ and the relative pressure $p=0$ is defined at $\left(x,y,z\right)=\left(4.8,1.3,0\right)$.

Figure 4 shows the time evolution of the velocity component $w$ at $\left(x,y,z\right)=\left(3.2,0.4,0\right)$ and $\left(3.2,0.4,5.2\right)$. Combined with the symmetry results in Figure 3 , the flow patterns are examined with respect to differences in spanwise boundary conditions and Reynolds number $Re$.

At $Re=300$, the flow is steady regardless of spanwise boundary conditions. Under wall boundary conditions, at $Re=350$, the flow is a symmetric three-dimensional oscillatory flow with $w=0$ in the $xy$-plane at $z=0$. At $Re=389$, the flow transitions to an asymmetric three-dimensional oscillatory flow with $w\ne 0$ at $z=0$. The dominant oscillatory component appears to be a simple sinusoidal wave but includes long-period undulations. At $Re=428$, the undulations intensify, and short-period irregular fluctuations emerge. These irregular oscillations become more pronounced at $Re=548$ and $Re=658$.

Under periodic boundary conditions, at $Re=350$, the waveform is a simple sinusoidal wave, and the flow remains two-dimensional with $w=0$ across all $z$. At $Re=389$, the oscillatory flow includes long-period undulations; $w=0$ at $z=0$, but oscillations with $w\ne 0$ are observed at $z=5.2$, indicating a symmetric three-dimensional oscillatory flow in the $xy$-plane at $z=0$. For $Re\ge 428$, $w\ne 0$ at $z=0$, and the flow transitions to an asymmetric three-dimensional oscillatory flow across all $xy$-planes, similar to the characteristics under wall boundary conditions.

These characteristics are confirmed to apply throughout the channel by visualizing vortex structures, as shown later in Figure 7 . For two-dimensional calculations, simple sinusoidal waves are obtained for all cases with $Re>300$.

Figure 5 illustrates the time evolution of the velocity component $v$ at $\left(x,y,z\right)=\left(3.2,0.4,0\right)$. The waveform shows a simple sinusoidal pattern at $Re=350$, long-period undulations at $Re=389$, and both long-period undulations and short-period irregular fluctuations at $Re\ge 428$.

Figure 6 shows the amplitude spectrum $\left|V\left(St\right)\right|$ of the velocity component $v$ at $\left(x,y,z\right)=\left(3.2,0.4,0\right)$. The time-series data shown in Figure 5 were analyzed for frequency characteristics using a fast Fourier transform (FFT). In previous two-dimensional simulations, $St=0.306$ was obtained regardless of $Re$, and experimental results for $Re<550$ showed $St$ values within the range of −10% to +5% relative to the two-dimensional simulation results, mostly concentrated in the range of −5% to 0% [5] . The oscillation characteristics were examined in conjunction with the time-series waveforms shown in Figure 5 .

First, the results for wall boundary conditions are discussed:

1) At $Re=350$, only the peaks of the fundamental wave and its harmonics were observed.

2) At $Re=389$, the amplitude spectrum exhibited a similar distribution to $Re=350$. However, a small peak appeared at $St=0.0488$ due to the long-period undulations shown in Figure 5 .

3) At $Re=428$, the peaks of the fundamental wave and its harmonics were clearly identifiable, but non-periodic components also emerged.

4) At $Re=548$, the proportion of non-periodic components further increased, although the peak of the fundamental wave remained distinguishable.

5) At $Re=658$, the peak of the fundamental wave could no longer be identified in the amplitude spectrum. The fundamental $St$ was $St=0.283$ for $Re=350$ to 428 and $St=0.273$ for $Re=548$.

Next, the results for periodic boundary conditions are discussed:

6) At $Re=350$, only the peaks of the fundamental wave and its harmonics were observed.

7) At $Re=389$, long-period undulations occurred, as shown in Figure 5 , but no peaks were observed in the amplitude spectrum.

8) At $Re=428$, similar trends to $Re=389$ were observed.

9) At $Re=548$, non-periodic components appeared alongside the fundamental wave and its first harmonic, whose peaks were distinguishable.

At $Re=658$, the distribution resembled that under wall boundary conditions, and the peak of the fundamental wave could no longer be identified in the amplitude spectrum. The fundamental $St$ values were $St=0.264$ for $Re=350$ and $Re=389$, $St=0.273$ for $Re=428$, and $St=0.283$ for $Re=548$.

Figure 7 shows an iso-surface of the second invariant of the velocity gradient tensor $Q$ at a given moment, with the flow moving from the top left to the bottom right. $Q$, is defined by Equation (6):

$Q=-\frac{1}{2}\frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i}\left(i,j=1,2,3\right).$ (6)

Regions where $Q>0$ are defined as vortex regions. The selection of $Q$-isosurfaces is arbitrary, and $Q=0$ is unsuitable for visualizing vortex structures as it encompasses most of the domain. In this study, the values were visually adjusted to ensure the enclosed volume by the isosurface accounted for 5% of the total volume, using the range $0.2\le Q\le 1.5$.

Common features across all conditions include the generation of elongated vortices in the spanwise direction, wrapping around the downstream corners. The maximum $Q$ values were observed at the vortex core, exceeding 1000 for $Re=658$. Results for wall boundary conditions are as follows:

1) At $Re=350$, vortices stretched thinly in the spanwise direction and were symmetric in the $xy$-plane at $z=0$.

2) At $Re=389$, similar thin, elongated vortices were observed, but they exhibited asymmetric vortex structures.

3) At $Re=428$, in addition to elongated vortices wrapping around the downstream corners, multiple vortices of varying sizes were identified.

At $Re=548$, these vortices became smaller and more numerous.

4) At $Re=658$, the vortices became further subdivided.

For periodic boundary conditions, the same $Q$-isosurfaces were used:

5) At $Re=350$, uniformly elongated vortices appeared in the spanwise direction.

6) At $Re=389$, spanwise symmetry was maintained in the $xy$-plane at $z=0$.

7) At $Re=428$, asymmetry developed in the spanwise direction, with localized vortex clusters.

8) At $Re=548$ and $Re=658$, vortex structures became further subdivided, similar to those under wall boundary conditions.

In summary, as $Re$ increases, the flow transitions as follows:

For both wall and periodic boundary conditions, the flow transitions to asymmetric three-dimensional oscillatory flow at $Re\ge 428$.

When oscillations involve low- $St$ undulations, the timing of the time-averaging operation affects the averaged values. Figure 8 shows the time-averaged velocity component $\langle u\rangle$ and the root mean square (RMS) of the fluctuating velocity component $u^{\prime }$ ( ${u^{\prime }}_{\text{RMS}}$) for every 100 dimensionless time units at $Re=658$. Data were collected at $\left(x,y,z\right)=\left(2.4,1.3,0\right)$, corresponding to the centerline in the $y$-direction of the ungrooved region of the channel. Model A was used for the mesh, and data were collected at intervals of 0.1 dimensionless time over 500 dimensionless time units. When the averaging range is $t=0$ to $T$, $\langle u\rangle$ and ${u^{\prime }}_{\text{RMS}}$ are defined by Equations (7) and (8):

$\langle u\rangle =\frac{1}{T}{\displaystyle {\int }_0^{T}u\text{d}t}$ (7)

${u^{\prime }}_{\text{RMS}}=\sqrt{\langle {u^{\prime }}^2\rangle }=\sqrt{\frac{1}{T}{\displaystyle {\int }_0^T{u^{\prime }}^2\text{d}t}}$ (8)

The same definitions were applied to velocity components $v$, $w$, and wall shear stress ${\tau }_0$ for time-averaging. The errors, assuming the median values as true values, were ±4.47% for $\langle u\rangle$ and ±14.0% for ${u^{\prime }}_{\text{RMS}}$. Similarly, errors for ${v^{\prime }}_{\text{RMS}}$ and ${w^{\prime }}_{\text{RMS}}$ were ±4.31% and ±5.69%, respectively. Subsequent discussions consider that the time-averaged velocity components and the RMS of fluctuating components include errors within these ranges. Unless otherwise noted, time-averaging was conducted using results over 100 dimensionless time units. In dimensional terms, based on the channel geometry and fluid properties used in the experiment [20] , 100 dimensionless time units correspond to 7.09 seconds at $Re=658$.

Figure 9 shows the distribution of $\langle u\rangle$ along the $y$-axis at $x=2.4$ and $z=0$. As seen from Figure 9(a) , at $Re=350$, $\langle u\rangle$ reached a maximum at $y=1.3$, the heightwise centerline of the ungrooved region. This maximum value is denoted as ${\langle u\rangle }_{\text{max}}$. Under wall boundary conditions using Model C, ${\langle u\rangle }_{\text{max}}=1.47$. Comparing errors among different meshes, ${\langle u\rangle }_{\text{max}}$ obtained with Model A and Model B showed errors of −1.09% and +0.31%, respectively, relative to Model C. Under periodic boundary conditions and two-dimensional conditions, ${\langle u\rangle }_{\text{max}}$ was −4.86% and −4.87%, respectively, compared to Model C. This indicates that the time-averaged maximum velocity in the main flow direction under wall boundary conditions is higher than that under periodic or two-dimensional conditions, consistent with previous reports for steady flows [7] . While at $Re=658$, as shown in Figure 9(b) , variations in the time-averaged components increased due to the timing of the averaging under both wall and periodic boundary conditions. The location of $y$ where ${\langle u\rangle }_{\text{max}}$ occurs became indeterminate but was confirmed to be within the range $1.25<y<1.45$ in this study. Qualitatively, at $Re=658$, the faster main flow was observed to sink into the grooved regions compared to $Re=350$.

Figure 10 shows the distribution of the time-averaged absolute wall shear stress $\langle \left|{\tau }_0\right|\rangle$. Data were collected at two points, A $\left(x,y,z\right)=\left(4.8,0.8,0\right)$ and B $\left(x,y,z\right)=\left(4.0,0.4,0\right)$, corresponding to nearly the same locations as in the experiment [3] . Both the simulation and experimental results are presented in dimensionless form in Figure 10 . Wall shear stress was non-dimensionalized as ${\tau }_0={\tau }_0^{*}/\left(\rho U^{*2}\right)$. The value of $\langle \left|{\tau }_0\right|\rangle$, like velocity, varied depending on the timing of the time-averaging operation. Using the same procedure as shown in Figure 8 , the variation of $\langle \left|{\tau }_0\right|\rangle$ for every 100 dimensionless time units was examined at $Re=658$ using Model A. The maximum and minimum values at point A were ${\langle \left|{\tau }_0\right|\rangle }_{\text{max}}=0.0153$ and ${\langle \left|{\tau }_0\right|\rangle }_{\text{min}}=0.0110$, respectively, while at point B, they were ${\langle \left|{\tau }_0\right|\rangle }_{\text{max}}=0.0188$ and ${\langle \left|{\tau }_0\right|\rangle }_{\text{min}}=0.0135$. The errors relative to the median values were ±10.2% at point A and ±7.9% at point B.

Figure 10. Time-averaged absolute value of dimensionless wall shear stress $\langle {\tau }_0\rangle$ as a function of the Reynolds number $Re$. The data were acquired at the two places: point A $\left(x,y,z\right)=\left(4.8,0.8,0\right)$ and point B $\left(x,y,z\right)=\left(4,0.4,0\right)$. Those points are the same places as the experiment [6] . (a) Point A, (b) Point B.

For $\langle \left|{\tau }_0\right|\rangle$ computed under the four conditions (Model A, Model B, Model C, and periodic boundary conditions), the errors relative to the median values at points A and B were ±26.6% and ±35.9% at $Re=428$, and a maximum of ±7.58% and ±12.9% for other $Re$. The discrepancy in $\langle \left|{\tau }_0\right|\rangle$ for $Re=428$ obtained with Model B compared to other meshes and boundary conditions is attributed to the occurrence of particularly large undulations by chance. As shown in the $Q$-isosurfaces in Figure 7 , localized vortex clusters emerge at $Re=428$, making the time-averaged values prone to greater variability. At $Re=548$, the fluctuation amplitude is significantly affected by low-frequency undulations, resulting in a greater sensitivity of time-averaged values to the duration and phase of sampling. This sensitivity amplifies the discrepancies among different mesh models.

When comparing the simulation results with experimental data, significant differences were observed at $Re=548$ in the channel center at $z=0$. However, no notable differences in $\langle \left|{\tau }_0\right|\rangle$ were observed between wall boundary conditions and periodic boundary conditions.

The time-averaged wall shear stress values at two representative points in the channel were compared with the experimental data from Nishimura et al. [6] . Overall, good agreement was observed across a wide range of Reynolds numbers, except at $Re=548$, where low-frequency undulations resulted in larger variability. These comparisons validate the numerical approach used in this study.