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TECHNICAL PAPERS

Analysis of Temporal Shear Stress Gradients During the Onset Phase of Flow Over a Backward-Facing Step

[+] Author and Article Information
Mark A. Haidekker, Charles R. White, John A. Frangos

University of California, San Diego, Department of Bioengineering, La Jolla, CA 92092-0412

J Biomech Eng 123(5), 455-463 (Apr 17, 2001) (9 pages) doi:10.1115/1.1389460 History: Received July 06, 2000; Revised April 17, 2001
Copyright © 2001 by ASME
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References

Figures

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Cell exposure to temporal gradients compared to cell proliferation rate. Shown is the cell proliferation of cells exposed to shear stress at sudden onset of flow (200 ms, –▴–) relative to the proliferation rate for ramped (15 s) onset of flow (–▪–) 27, lower panel. The difference between cell proliferation rates with sudden and ramped onset of flow are statistically significant within the fields of view from 2.25 to 7.88 mm (shaded area). The total exposure of the cells to temporal shear stress gradients is shown in the middle panel for sudden onset of flow (solid line) versus ramped onset of flow (dashed line). Similarly, the maximum exposure of the cells to spatial gradients is shown in the upper panel for both flow conditions.
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Immunofluorescent micrographs of BrdU positive nuclei after 4 h exposure to recirculating flow in the backward-facing step chamber. Shown is a representative example from a paired set of HUVEC slides with cells exposed to a sudden onset of flow in the upper panel, and cells exposed to ramped flow (15 s) below. Each micrograph is a photo-construct that comprises three contiguous HPF. Micrographs span the region of flow recirculation and reattachment (2.2 mm to 9.0 mm downstream from the expansion site). The white dots represent BrdU positive proliferating nuclei. The scaling bar equals 1.1 mm, and the arrow indicates the position of the second stagnation point.
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Dependency of the maximum shear stress in the recirculation zone (□) and the forward reattachment zone (♦) on the expansion ratio. A high expansion ratio leads to an amplification of the maxima relative to the outflow shear stress.
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The maximum spatial gradient of shear stress as a function of the ramp time. The data points were computed for the same discrete ramp times as in Fig. 8. A good fit can be achieved with an exponential decay function, τmax=s⋅exp(−tramp/Kt)+P with the constants s=42.7 kPa/m,Kt=8.6 ms, and P=40.11 kPa/m and a regression coefficient of R=0.996. As opposed to temporal gradients of shear stress, spatial gradients do not vanish with slow ramping times, but rather approach a lower boundary. Also, the increase of spatial gradients with shorter ramp times is by orders of magnitude less pronounced than in the case of temporal gradients.
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The maximum temporal gradient of shear stress as a function of the ramp time. The data points were computed for the discrete ramp times 50, 100, 200, 500 ms and 1, 2, and 15 s. Due to scaling reasons, the 15 s data point is not shown. A nonlinear curve fit shows that the data points are well represented by the function τ̇max=A⋅tramp−B with A=221.5 and B=1.15(R2=0.999). With tramp→∞ the temporal gradient τ̇ vanishes.
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Spatial partial derivative of the function shown in Fig. 5. As in Fig. 5, the same data are presented as a three-dimensional graph and a two-dimensional false-color contour plot. It can be seen that the zone with the highest spatial gradients matches the second stagnation point. At the end of the ramping period, a maximum spatial gradient can be observed that is slightly higher than the steady-state gradient. As opposed to the temporal gradients, the spatial gradients do not vanish when the system reaches equilibrium.
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Temporal partial derivative of the function shown in Fig. 5. As in Fig. 5, the same data are presented as a three-dimensional graph and a two-dimensional false-color contour plot. The graphs show that the highest temporal gradient of the shear stress occurs at the time the ramping ends. The system then reaches a steady state after approximately 300 to 350 ms, after which the temporal gradient remains zero. The zone exposed to the highest temporal gradients is located slightly upstream of the second stagnation point.
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Graphs of the bottom plate shear stress as a function of location and time, τ(x,t). The same function is shown as a three-dimensional graph (A) and a false-color contour plot (B) with yellow and red colors indicating the forward shear stress after the second stagnation point and blue colors indicating the recirculation zone. The final position of the second stagnation point is indicated by the white arrow. The onset phase of flow can easily be recognized by the rise of shear stress and the movement of the second stagnation point: Moving forward in time is equivalent to moving to the right. The sudden-expansion site is represented by the lower edge of the right graph or the right edge moving towards the right background in the left graph. The flow direction is indicated by black arrows. [Subscribers to the JOURNAL OF B IOMECHANICAL E NGINEERING may access Figs. 5(B), 6(B), and 7(B) in color on-line at http://ojps.aip.org/ASMEJournals/Biomechanical.]
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Development of the recirculation eddy in dependency of time. The Reynolds number is ramped linearly over 200 ms (dashed line). After 200 ms, when the inflow velocity is kept constant, the recirculation zone keeps increasing in size. At 300 ms, the position of the second stagnation point lies within 2 percent of its steady-state value, X/S=8.6, which corresponds to 4.8 mm downstream of the expansion site.
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Streamlines of the developing flow in the flow chamber during a simulated 200 ms linear onset phase of the inflow velocity. After 200 ms, the inflow velocity reaches its final values of 3.5 ml/s. A steady state is reached approximately 300 ms after the initial onset of flow. During the final 100 ms, the recirculation eddy keeps developing, although the flow rate is constant.
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Upper panel: Calculated changes of the highest shear stress values with increasing grid size. Resolutions of 70×14,150×28,300×50,600×100, and 1200×200 grid nodes were chosen, and the steady-state flow at Re=243 computed. There is no significant change (less than 3 percent) between the two largest grid sizes. Lower panel: Different time step sizes (parameter κ in Eq. (4)) were used to verify the convergence of the discretization of the momentum equation. No significant changes were observed for κ≤0.6.
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Geometry of the backward-facing step flow chamber with the dimensions used in the computer simulation. The origin of the coordinate system is the lower right corner of the expansion step. In order to keep the grid size small, only the first 15 mm downstream of the expansion site were considered in the model, since the fully reattached parabolic-profile flow is established at that point. Also shown are the streamlines of the steady-state flow at Re=243 with the second stagnation point at 4.8 mm.

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