# Quasi-Periodic Oscillating Flows in a Channel with a Suddenly Expanded Section

^{*}

## Abstract

**:**

## 1. Introduction

- Stability: When a small disturbance is added to a steady flow, if any disturbance is attenuated, the flow is “stable”. Conversely, if one of any disturbance grows, its state is “unstable”.
- Turbulence: In this paper, irregularly changing flow is treated as “turbulent flow”, but the definition of the turbulent flow has not always been obvious. Here is a summary of the characteristics of turbulent flow. The physical quantities such as velocity and static pressure fluctuate irregularly in time and space, and the transport and exchange are much faster than the molecular transport. The energy of fluctuation is transferred from usual large-scaled vortices to small-scaled ones.
- Transition: A process in which the flow changes from a stable state to an unstable state as the Reynolds number increases is called a “transition”. The forms of transition process differ depending on the type of the base flow. In the case of a plane Poiseuille flow, the oscillatory flow is quasi-periodic just above the critical Reynolds number.

_{c}in this aspect ratio and boundary condition was reported as Re

_{c}= 843 [8] or 1000 < Re

_{c}< 1100 [19]. Therefore, in this study, the computations were carried out for the range of Reynolds numbers 1200 ≤ Re ≤ 2000, where the flow should exhibit oscillatory flow, keeping a certain numerical accuracy.

## 2. Methods

#### 2.1. Geometry of a Channel and Governing Equations

_{0}, y

_{0}). Each length of the sides is shown in Figure 1. Parameters describing the shape of the channel are the expansion ratio Ex = (y

_{3}− y

_{−3})/(y

_{1}− y

_{−1}) = 3 and the aspect ratio As = (x

_{1}− x

_{−1})/(y

_{3}− y

_{−3}) = 7/3.

**u**= (u, v) are numerically obtained as a function of time t and

**x**= (x, y). Dimensionless governing equations are the Equation of continuity (1) and the Navier–Stokes Equation (2). In this study, viscous dissipation is ignored, and therefore, the energy equation is not taken into account.

^{2}/s] and density ρ* [kg/m

^{3}] are assumed to be constant. Representing dimension values with the star (*), the physical values are made dimensionless as t = t*U*/δ*,

**x**=

**x***/δ*, p = p*/ρ*U*

^{2}, and

**u**=

**u***/U*.

#### 2.2. Numerical Solutions

**u**were calculated by the PISO (Pressure Implicit with Splitting of Operators) algorithm. The numerical schemes were constructed with the second-order accuracy in all, where the second-order accurate Crank–Nicholson scheme for the time t and the second-order accurate central-difference scheme for the space

**x**were used. The pressure p and the velocity

**u**are solved by the GAMG (geometric–algebraic multi-grid) method and Gauss–Seidel method, respectively. The iterative calculation was performed until the residual value, which was defined by normalizing the L

_{1}norm of the residual vector [24] until it became less than 10

^{−10}.

**u**is given by the parabolic distribution as the exact solution of the plane Poiseuille flow, and the pressure p is the zero gradient in the streamwise direction, as shown in Equation (3).

**u**, the Sommerfeld radiation condition is used, and p is set so that the mean on the outlet is zero, as shown in Equation (4). Equation (4) is the advection equation about

**u**. It means that the velocity upstream by U

_{c}Δt in the x direction from the boundary surface at a current time step is applied to the velocity on the boundary surface at the next time step, where Δt is the time step. For the advection velocity U

_{c}of the Sommerfeld radiation condition, the area-averaged velocity is applied at each boundary surface subdivided by a mesh.

**u**, and the zero gradient in the normal direction is used for p, as shown in Equation (5).

_{0}≤ x ≤ x

_{2}in the mainstream x direction, as well as in the y direction such as in a turbulent channel flow. In Figure 1, the thin lines in the computational region indicate the boundary lines through which the direction of coarseness and fineness is exchanged.

_{x}(the number of the meshes in the x direction) × N

_{y}(the number of the meshes in the y direction). The intervals of the unequal interval meshes vary in the geometric progression so that the ratio of the maximum and the minimum of them is R

_{x}= R

_{y}= 10. The verification of accuracy of the discretization is checked by comparing the three kinds of the meshes, which is described in Section 2.3. The following shows the results of Model C, unless otherwise specified.

_{max}was 0.3 or less by considering the numerical simulation of the plane Poiseuille flow [2]. The time step in Model C is 0.002 (Δt = 0.002), and the time step in the others is adjusted so that Co

_{max}becomes almost equal to Model C. Details of the meshes are shown in Table 1. Mainly using an Intel Xeon E3-1241 v3 (3.50 GHz × 8) for CPUs, the required time for the calculation of Model C was 12.4 days for 2000 dimensionless times at Re = 2000 with six threads paralleled.

#### 2.3. Evaluation of Discretization Error

_{max}of the time-fluctuating velocity component v′ was selected as the targeted physical value, which was sampled by the position at (x, y) = (10, 0.5). As a result, v′

_{max}was 0.0666, 0.0979, and 0.110 for Δx = 1/20, 1/40, and 1/80, respectively. Based on them, the scheme order was calculated as o = 1.34 by Equation (6), where r = 2, φ = v′

_{max}, and l = Δx = 1/80 were substituted. The amplitude v′

_{max}converges to a constant value, so the characteristic of the target can be reproduced enough.

_{max}was 0.0990, 0.0979, and 0.0974 for Δt = 0.01, 0.005, and 0.0025, respectively. The scheme order was calculated as o = 1.32.

_{1}and its amplitude |V(f

_{1})| of the velocity component v at (x, y) = (10, 0.5) were calculated by the discrete Fourier transform (DFT). The program of the DFT is supplied as the application [26]. When the equal interval mesh system (Model A) was used, the flow changed to chaos as soon as the computation was started at Re = 2000. On the other hand, a quasi-periodic oscillatory flow was able to be obtained with the unequal interval meshes at Re = 2000. Specifically, the results on Model C were f

_{1}= 0.0792 and |V(f

_{1})| = 0.0652. Even on Model B, which is courser than Model C, and whose results f

_{1}= 0.0785 and |V(f

_{1})| = 0.0606 are almost equal to those on Model C. Therefore, it is considered that the mesh of Model C has enough accuracy of discretization.

## 3. Results and Discussion

#### 3.1. Periodic Characteristics

^{−2}; thus, it suggests that the flow transition to a quasi-periodic state having the waveform of the modulated wave takes place. In the range of Re ≥ 1600, it exhibits irregular quasi-periodic flow, as recognized from the mixing of the non-harmonic, which includes the aperiodic components for the fundamental wave. The non-harmonic increases with the Reynolds number, and at Figure 4c, where Re = 1600, only the weak non-harmonic is discretely mixed, but at Figure 4d, where Re = 2000, the non-harmonic is mixed continuously.

#### 3.2. Vortex Generation

_{0}was calculated. Paying attention to the flat plate at y = 1 in the downstream region of the sudden contraction part, τ

_{0}is defined as Equation (7), where τ

_{0}< 0 for the region where there is no vortex. If the three vortices happen, the sign of τ

_{0}changes twice from minus to plus when the group of the vortices passes at a sampling point.

_{0}for t = 1000–2000. The magnitude of τ

_{0}correlates with the size of the vortex. The maximum at Re = 2000 is clearly larger than one in Re ≤ 1800 because the much bigger groups of the vortices have been generated several times during the computation.

_{1}≈ 0.079. Most of the vortices disappear just behind the sudden contraction part (x ≈ 7) as soon as they are generated, but the vortices are generated again in the downstream region, and the number of them is at a maximum near x ≈ 18. Note that the sign reversal occurs more than 70 times at x = 7, since a part of the main flow passing through the sudden contraction part flows backward, and it joins up with the circulation vortex in the upstream region of the expanded section when the flow pattern of the stream line changes from a crest to a trough.

#### 3.3. Effect of a Sudden Contraction Part

_{c}varied with the reduction ratio Cn, and such values were Re

_{c}= 2310 (Cn = 2), 1013 (Cn = 4), and 825 (Cn = 8). Judging from this tendency, the critical Reynolds number at Cn = 3 was expected to be 1013 < Re

_{c}< 2310. The present time-marching computations implemented using a Model E mesh system gave an asymmetric steady flow at Re = 2000 and an irregular unsteady flow at Re = 4000. The channel flow with the sudden contraction part immediately changed from a steady state to a time-dependent irregular state as the Reynolds number increases. Although Model F was also carried out for Re = 4000, it took a very long computational time; therefore, we just confirmed that there was no significant difference in the vortex generation pattern between Model E and Model F.

#### 3.4. Turbulent Kinetic Energy

_{H}, v′

_{H}and the non-harmonics u′

_{N}, v′

_{N}, as shown in Equation (8) and Figure 11.

_{i}= i/(NΔt) of the number i based on the sampling period Δt = 0.5, the amplitude a

_{i}and the phase θ

_{i}are obtained as 1/N times the absolute value and the argument of the complex numbers obtained with the DFT. Then, assuming that the components of the m-th harmonic (m = 1, 2, …) are f

_{m}, a

_{m}and θ

_{m}, the harmonic component u′

_{H}is defined by Equation (10).

_{c}considering the Nyquist frequency f

_{c}= 1/(2Δt). Moreover, the relation that cos(2πf

_{i}t + θ

_{i}) = cos(2πf

_{N−i}t + θ

_{N−i}) is valid by the symmetries for the Nyquist frequency of the DFT and the trigonometric function, so each harmonic component is doubled.

_{H}, and a non-harmonic component k

_{N}are defined as shown in Equations (11)–(13), where $\overline{{{u}^{\prime}}^{2}}$, $\overline{{{v}^{\prime}}^{2}}$, $\overline{{{u}^{\prime}}_{H}{}^{2}}$, $\overline{{{v}^{\prime}}_{H}{}^{2}}$, $\overline{{{u}^{\prime}}_{N}{}^{2}}$, $\overline{{{v}^{\prime}}_{N}{}^{2}}$ are obtained by the arithmetic mean of the time-fluctuating velocity component.

_{H}, and k

_{N}. At Re = 1400, the flow is almost periodic, having the waveform of the modulated wave, so the patterns of three components are similar to each other. When the flow is quasi-periodic as shown in the cases at Re = 1600 and Re = 2000, the patterns are different between the harmonics k

_{H}and the non-harmonics k

_{N}, but the position of a peak does not tend to be changed irrespective of the Reynolds number.

_{N}is less than 1% compared with that of k

_{H}at Re = 1400, but both have the same order maximum value for Re = 1600. At Re = 2000, the maximum value of k

_{N}is larger than that of k

_{H}.

_{H}is dominant. Its fundamental wave is the antisymmetric disturbance, because the velocity component u is an odd function and v is an even function. Therefore, the maximum value of the peak does not exist at y = 0, which is the center line of the channel, but it exists at y ≈ 0.5.

_{H}.

_{N}unlike the cases mentioned above. Therefore, it can be seen that the aperiodicity of the flow is generated not in the downstream of the sudden contraction part but in the expanded section on the upstream side of the corner. Although the region where k

_{N}becomes apparent is limited, k

_{N}spreads slightly to the downstream, and it appears clearly that the change of the vortices size is observed just after the sudden contraction.

_{N}near the contact of the vortices in the expanded section. That is, it is considered that the flow aperiodicity is an inherent property for the expanded section, and it is caused by the interaction of the circular vortices at a certain Reynolds number or more. The circular vortex hardly transports the non-harmonic component from the sudden contraction part. In addition, no conspicuous peak has been observed in the non-harmonic component of the turbulent kinetic energy in the downstream of the sudden contraction part, so the separation vortex does not significantly affect the aperiodic oscillating flow. Therefore, the aperiodicity of flow and the generation of vortices occur independently, although they are interacted with each other. Since the generation of the separation vortices did not always occur, it would be a secondary phenomenon that occurs irregularly.

## 4. Conclusions

- At the Reynolds number Re = 1200, it is the periodic oscillating flow. At Re = 1400, it becomes the quasi-periodic oscillating flow having the waveform of the modulated wave, which includes the low-frequency components. In Re ≥ 1600, it is the quasi-periodic oscillating flow with the disordered fluctuations.
- For Re ≥ 1600, the separation vortices are sometimes generated because the wave crest caused by the oscillating main flow cannot keep touching the wall before it passes the corner of the sudden contraction part. Furthermore, the size of the vortex is not constant, and the vortex is not always generated. Most of the vortices disappear just behind the sudden contraction, but the vortex appears again in the downstream region.
- When the turbulent kinetic energy is decomposed into the harmonic component and the non-harmonic component, the peaks of the non-harmonic component are localized near the contact of the vortices in the expanded section. That is, the aperiodic characteristics of the oscillating flow are generated in the expanded section. The circular vortex hardly transports the non-harmonic component from the sudden contraction part.
- The aperiodicity of flow and the generation of vortices interact with each other, but each phenomena occurs independently. It has been exhibited that the aperiodicity of flow is hardly affected by vortex generation, since the peak of the non-harmonic component of turbulent energy exists in the expanded region.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

As | aspect ratio [-] |

Cn | contraction ratio [-] |

Co_{max} | maximum courant number [-] |

Ex | expansion ratio [-] |

f | dimensionless frequency = f*δ*/U* [-] |

f* | frequency [Hz] |

f_{1} | dimensionless fundamental frequency [-] |

k | dimensionless turbulent kinetic energy [-] |

N | number of samples on discrete Fourier transform [-] |

N_{x} | number of meshes in the x direction [-] |

N_{y} | number of meshes in the y direction [-] |

p | dimensionless pressure = p*/ρ*U*^{2} [-] |

p* | pressure [Pa] |

R_{x} | ratio of the maximum and the minimum of meshes in the x direction [-] |

R_{y} | ratio of the maximum and the minimum of meshes in the y direction [-] |

Re | Reynolds number = U*δ*/ν* [-] |

Re_{c} | critical Reynolds number [-] |

t | dimensionless time = t*U*/δ* [-] |

t* | time [s] |

u | dimensionless velocity vector = (u, v) [-] |

u | dimensionless x-directional velocity component = u*/U* [-] |

u* | x-directional velocity component [m/s] |

U* | characteristic velocity as the maximum of the plane Poiseuille flow [m/s] |

U_{c} | advection velocity [-] |

v | dimensionless y-directional velocity component = v*/U* [-] |

v* | y-directional velocity component [m/s] |

v′ | dimensionless time-fluctuating velocity component of v [-] |

v′_{max} | dimensionless maximum amplitude of v′ [-] |

|V(f)| | dimensionless amplitude spectrum of v [-] |

x | dimensionless coordinate vector = (x, y) [-] |

x | dimensionless x coordinate = x*/δ* [-] |

x* | x coordinate [m] |

y | dimensionless y coordinate = y*/δ* [-] |

y* | y coordinate [m] |

Greek Symbols | |

Δf | dimensionless frequency interval on discrete Fourier transform [-] |

Δt | dimensionless time step [-] |

Δx | dimensionless mesh size in the x direction [-] |

Δy | dimensionless mesh size in the y direction [-] |

δ* | characteristic length as the half height of the channel [m] |

ν* | coefficient of kinetic viscosity [m^{2}/s] |

ρ* | density [kg/m^{3}] |

τ_{0} | dimensionless wall shear stress on the upper wall = τ_{0}*/ρ*U*^{2} [-] |

τ_{0}* | wall shear stress on the upper wall [Pa] |

Subscripts or Superscripts | |

* | dimension values |

¯ | time-mean |

′ | time-fluctuating |

_{H} | harmonics (periodic components) |

_{N} | non-harmonics (aperiodic components) |

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**Figure 2.**Mesh structure of the finest mesh, Model C. This picture shows the range of 6 ≤ x ≤ 8 and 0.8 ≤ y ≤ 1.2.

**Figure 3.**Time evolution of the velocity component v at (x, y) = (10, 0.5). (

**a**) It was a periodic oscillating flow. At Re = 1400, as shown in (

**b**), the local maximum is not constant, so it is a quasi-periodic oscillating flow with a modulated wave containing low frequency components. At Re = 1600, as shown in (

**c**), in addition to having a waveform of a modulated wave, it is a quasi-periodic oscillating flow with irregular fluctuations. At Re = 2000, as shown in (

**d**), the irregularity becomes even stronger.

**Figure 4.**Amplitude spectrum |V(f)| of the velocity component v at (x, y) = (10, 0.5). (

**a**) Re = 1200. (

**b**) Re = 1400. (

**c**) Re = 1600. (

**d**) Re = 2000.

**Figure 5.**Stream lines at Re = 2000, which represent the evolution process of a group of vortices. (

**a**) t = 1531. (

**b**) t = 1533. (

**c**) t = 1538.

**Figure 8.**Mesh structure of the non-matching mesh for Model E. This picture shows the range of −5 ≤ x ≤ 10 and 0 ≤ y ≤ 3.

**Figure 9.**Stream lines in the channel with the sudden contraction part at Re = 4000 at a certain moment.

**Figure 10.**Amplitude spectrum |V(f)| of the velocity component v in the channel with the sudden contraction part at (x, y) = (10, 0.5) at Re = 4000.

**Figure 11.**The time-fluctuating velocity component v′, which was decomposed into the harmonic v′

_{H}and the non-harmonic v′

_{N}at (x, y) = (10, 0.5). (

**a**) Fluctuation. (

**b**) Harmonic. (

**c**) Non-harmonic.

**Figure 12.**Distributions of the turbulent kinetic energy k, which is decomposed into the harmonic component k

_{H}and the non-harmonic component k

_{N}.

Name | N_{x} × N_{y} | R_{x}, R_{y} | Δt | ||
---|---|---|---|---|---|

−20 ≤ x ≤ −7 | −7 ≤ x ≤ 7 | 7 ≤ x ≤ 40 | |||

Model A | 260 × 40 | 280 × 120 | 660 × 40 | 1 | 0.005 |

Model B | 117 × 46 | 224 × 138 | 479 × 46 | 10 | 0.0025 |

Model C | 195 × 76 | 373 × 228 | 798 × 76 | 10 | 0.002 |

Name | N_{x} × N_{y} | R_{x}, R_{y} | Δt | |||
---|---|---|---|---|---|---|

−20 ≤ x ≤ 0 | 0 ≤ x ≤ 7 | 7 ≤ x ≤ 20 | 20 ≤ x ≤ 40 | |||

Model D | 180 × 138 | 161 × 138 | 299 × 46 | 180 × 46 | 10 | 0.0025 |

Model E | 100 × 76 | 268 × 228 | 498 × 76 | 300 × 76 | 10 | 0.002 |

Model F | 100 × 76 | 451 × 384 | 838 × 128 | 300 × 76 | 10 | 0.001 |

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**MDPI and ACS Style**

Masuda, T.; Tagawa, T.
Quasi-Periodic Oscillating Flows in a Channel with a Suddenly Expanded Section. *Symmetry* **2019**, *11*, 1403.
https://doi.org/10.3390/sym11111403

**AMA Style**

Masuda T, Tagawa T.
Quasi-Periodic Oscillating Flows in a Channel with a Suddenly Expanded Section. *Symmetry*. 2019; 11(11):1403.
https://doi.org/10.3390/sym11111403

**Chicago/Turabian Style**

Masuda, Takuya, and Toshio Tagawa.
2019. "Quasi-Periodic Oscillating Flows in a Channel with a Suddenly Expanded Section" *Symmetry* 11, no. 11: 1403.
https://doi.org/10.3390/sym11111403