^{1}

^{1}

^{2}

^{2}

^{1}

^{2}

The flow instability through the side branch of a T-junction is analyzed in a numerical simulation. In a previous experimental study, the authors clarified the mechanism of fluid-induced vibration in the side branch of the T-junction in laminar steady flow through the trunk. However, in that approach there were restrictions with respect to extracting details of flow behavior such as the flow instability and the distribution of wall shear stress along the wall. Here the spatial growth of the velocity perturbation at the upstream boundary of the side branch is investigated. The simulation result indicates that a periodic velocity fluctuation introduced at the upstream boundary is amplified downstream, in good agreement with experimental result. The fluctuation in wall shear stress because of the flow instability shows local extrema in both the near and distal walls. From the numerical simulation, the downstream fluid oscillation under a typical condition has a Strouhal number of 1.05, which approximately agrees with the value obtained in experiments. Therefore, this periodic oscillation motion is a universal phenomenon in the side branch of a T-junction.

T-junction is a typical fluid pipe element used in the construction of power plants. Over the past three decades, many experimental and numerical researchers have investigated laminar flow inside T-junction for relevance in hemodynamics [

More recently, the focus in the hemodynamics field has been the flow instabilities induced in a cerebral aneurysm from the perspective of their growth and rupture [

Brown & Roshko [

In previous studies of the T-junction [

However, the experimental approach was unable to clarify in detail how the mechanism of the periodic oscillation occurs, i.e. the spatial amplification of flow instability and the distribution of wall shear stress (WSS). In the present study, this mechanism is clarified numerically. The mechanism of periodic oscillation induced in the side branch obtained in the experiment was verified in numerical simulation.

Numerical simulations were performed of flow in a T-junction (

universality of the independency for the Strouhal number, a typical setting employed a Reynolds number of Re_{S} = 700 (Q_{S}/Q_{T} = 0.50) in the side branch corresponding to Re_{T} = 800 in the trunk. Also, the tube radius ratio and the flow division ratio were fixed at R_{S}/R_{T} = 0.57 and Q_{S}/Q_{T} = 0.25, 0.33 and 0.5 [_{S}/Q_{T} = 0.5 because the oscillation frequency is clear at this condition in experiment. Furthermore, in numerical calculation, we examined the spatial amplification of velocity due to flow instability and the distribution of wall shear stress.

The velocity components at 480 numerical grid points in the upper half plane (grid B in _{3} in _{i} in Cartesian coordinate.

where d_{i} is each distance from the grid in cylindrical coordinate to the surrounding grids in Cartesian coordinate. The numerical domain is L = 0.046 m,

grid A | grid B | |
---|---|---|

Grid points N_{r} × N_{θ} × N_{z} | 10 × 24 × 24 | 20 × 48 × 132 |

Grid spacing h_{r} × h_{θ} × h_{z} | ||

Time step Δt | 0.1 | 0.01 |

Total residual at convergence ε | 0.1 × 10^{−3} | 0.1 × 10^{−1} |

Reynolds number Re_{S} | 700 | 700 - 4000 |

i.e. X/R_{S} = 2.89 - 9.46.

A cylindrical coordinate system (r, θ, z) was specified for the side branch (

and the continuity equation,

where the dimensional variables (u, v, w), (r, θ, z) and (X, Y, Z) are normalized using mean velocity U_{S} = 8 ´ 10^{−2} m/s and radius of R_{S} = 7 ´ 10^{−}^{3} m in the side branch. Here, the kinematic viscosity and the density of the working fluid are ν = 1.58 ´ 10^{−6} m^{2}/s and ρ = 1830 kg/m^{3}, respectively.

In the following numerical simulation, the steady flow is first obtained, and then an unsteady flow simulation was performed with the upstream boundary condition, in which the velocity field rotates around the axis of the side branch with a small angle around the axis of the side branch.

As depicted in

The amplitude of the rotation angle Δθ_{0} is set to π/180 (1˚), and the frequency equals the empirical value observed in the experiment, St_{S} = 1.03 (f = 6 Hz) [

The free-stream condition is applied at the downstream boundary. Thus, the flow instability in the side branch was generated numerically as follows. The time-dependent simulation was performed with the initial condition until a steady oscillating flow was obtained.

The discretized representations of the relevant equations were obtained through the control volume method, and solved using an iterative algorithm employed in the original code similar to SIMPLER method developed by Patanker [

The numerical settings are listed in

The amplification degree is defined as the ratio of (U_{rms})_{max} within X/R_{S} = 2.89 − 9.46 to (U_{rms})_{0} at the section S_{3} as follows,

where T and A are period and cross sectional area of side branch, respectively. When various frequencies, specifically, 0.1 Hz and 1 Hz to 12 Hz in steps of 1Hz, were set for the perturbation at the upstream boundary section, the degree of amplification in the spatial change of axial velocity across each cross section examined downstream in the side branch depended on the given frequency. When the ratio of the amplification degree (U_{rms})_{max}of at each section to (U_{rms})_{0} at section S_{3} converged within 5% of the maximum amplification degree found under various frequencies as described above, the given frequency was regarded to be reasonable. This is the major criterion for the convergence in the present numerical simulation.

A time dependent simulation was performed with a null velocity field as an initial condition. After the transient flow diminished, the flow solution converged onto a steady state, i.e. the flow oscillation appearing in experiments was not reproduced in this simulation. The fixed upstream velocity boundary condition has probably stabilized the flow in the side branch. For this reason, the effect of a small perturbation given in (4) was examined in relation to the stability of the flow field.

Under the periodic oscillations imposed on the upstream boundary condition as described above, the oscillation in the axial velocity component at the off-axis point (X/R_{S}, Z/R_{S}) = (0, 0.25) for three axial sections S_{3}, S_{5} and S_{7} (X/R_{S} = 2.89, 4.32, 5.75) was shown in _{3} and a little large in section S_{5}. It becomes much larger in section S_{7}. Further downstream of section S_{7}, the amplification attenuates asymptotically. Thus, the spatial amplification and section of velocity relates to the flow instability. The small periodic oscillation given at the upstream boundary S_{3} is substantially amplified at the downstream location S_{7 }with a phase lag of 0.21π rad which is calculated from the time difference Δt = 0.2, i.e. t = 12.7 and t = 12.9, of the minimum velocity in sections S_{3} and S_{7}, respectively. As one period is 1.9, the phase lag is 2π × 0.2/1.9 = 0.21π. This one period of 1.9 corresponds to the periodic frequency of f = 6.00 Hz, i.e. St_{S} = 1.05, which agrees well with the experimental results of 1.03 [

Contours plots of the secondary velocity vector and the iso-axial velocity contours at the downstream sections S_{5} and S_{7} (X/R_{S} = 4.32 and 5.75) at t = 12.7 (_{5} (X/R_{S} = 4.32), which are also seen in the upstream boundary, moving toward the near wall while another pair of slightly stronger vortices grew larger; the iso-axial velocity contours are a little asymmetric. Section S_{7}(X/R_{S} = 5.75) further downstream features two pairs of small and large vortices merging to form one pair of large vortices. Significant asymmetry is indicated in the axial velocity distribution; also, the iso-axial velocity contour is asymmetric because of flow instability, i.e. the small periodic velocity oscillation introduced in the upstream boundary has amplified downstream. The current simulation results at the sections S_{5} and S_{7} qualitatively agree with the experimental results (_{S} = 3.17 immediately downstream

from section S_{3} [^{−1}) at this layer is 6 times larger than that (47 s^{−1}) at the tube wall downstream of the side branch; the tangential velocity profile indicates a clear inflection point. This is one feature associated with the flow instability in the side branch of T-junction, which is one kind of Kelvin-Helmholtz instabilities. In the upper half of

The range of Strouhal number St_{S}, i.e. oscillating frequency, in the present simulation approximately agrees with that obtained in experiment (_{S} = 0.9 - 1.1, although there is a little deviation between the simulation and the experiment. Indeed, the observed oscillation depends on the amplification of flow instability in the side branch.

The time average WSS τ_{uw} in the distal wall in _{uw} in the section S_{3} is globally larger than that in the downstream region. The root mean square (RMS) of WSS τ_{uw-rms} = 0.9 at X/R_{S} = 8.5 in the downstream region as shown in _{S} = 3.2 downstream of section S_{3}. This difference in τ_{uw-rms} is closely related to the flow instability that is amplified in the downstream region. Furthermore, the amplitude of the resultant WSS τ_{uw-amp} = 1.75 at the near wall X/R_{S} = 6.5 denoted by A and τ_{uw-amp} = 2.50 at the distal wall X/R_{S} = 8.5 denoted by B are large in

The mechanism for flow instability in the side branch of a T-Junction in steady laminar flow was clarified using a periodic velocity perturbation introduced as

an upstream boundary condition in a numerical simulation. The simulation result qualitatively agrees with experimental measurements. The spatial amplification accompanied with the flow instability in the side branch by numerical simulation explains the occurrence of the velocity oscillation observed in the experiment. Therefore, this periodic oscillation motion is a universal phenomenon in the side branch of a T-junction.

The authors thank Mr. Takashi Fujiwarain Chiba University for suggestions concerning data processing.

In this study, the authors cited

The authors declare no conflicts of interest.

Yamaguchi, R., Tanaka, G., Nakagawa, T., Shirai, A., Liu, H. and Hayase, T. (2017) Universality of Periodic Oscillation Induced in Side Branch of a T-Junction in Numerical Simulation. Journal of Flow Control, Measurement & Visualization, 5, 73-85. https://doi.org/10.4236/jfcmv.2017.54006

f: frequency of fluid vibration

L: streamwise length of simulation domain

n: coordinate normal to shearing separation layer

Q_{S}: flow rate through side branch

Q_{T}: flow rate through trunk

q: tangential velocity along shearing separation layer

Re_{S} = 2R_{S}U_{S}/ν: Reynolds number based on variables of side branch

Re_{T} = 2R_{T}U_{T}/ν: Reynolds number based on variables of trunk

R_{S}: radius of side branch= 7 mm

R_{T}: radius of trunk= 12.25 mm

(r, θ, z): cylindrical coordinate defined in side branch

St_{S} = 2fR_{S}/U_{S}: Strouhal number based on variables of side branch

t= t^{*}/(R_{S}/U_{S}): dimensionless time

t^{*}: dimensional time

U_{rms}: root mean square root of axial velocity

(U_{rms})_{0}: root mean square root of axial velocity at section S_{3}

U_{S}: mean velocity in side branch

U_{T}: mean velocity in trunk

(u, ν, w): velocity component in (r, θ, z) coordinate direction, respectively

(u', ν', w'): velocity fluctuation of each velocity

(X, Y, Z): Cartesian coordinate defined in the whole right angle branch

τ_{uw}: non-dimensional resultant wall shear stress [= τ_{uw}^{*}/(4ρνU_{T}/R_{T})]

ν: kinematic viscosity of working fluid

ρ: density of working fluid

Submit or recommend next manuscript to SCIRP and we will provide best service for you:

Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.

A wide selection of journals (inclusive of 9 subjects, more than 200 journals)

Providing 24-hour high-quality service

User-friendly online submission system

Fair and swift peer-review system

Efficient typesetting and proofreading procedure

Display of the result of downloads and visits, as well as the number of cited articles

Maximum dissemination of your research work

Submit your manuscript at: http://papersubmission.scirp.org/

Or contact jfcmv@scirp.org