Viscous Incompressible Flow in Over a Step 2009
Computational Fluid Dynamics
Project 4
Viscous Incompressible Flow Over a Step
3/5/09
Shiva Naraharisetty
Robbie Driscoll
Sandeep Kumar
William Stoddard
Rajiv Kattekola
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Viscous Incompressible Flow in Over a Step 2009
Table of Contents
Problem formulation
3
Gambit and FLUENT software
5
Results
6
Case 1: Re 300 steady
7
Case 2: Re 300 unsteady
11
Case 3: Re 507 steady
14
Case 4: Re 507 unsteady
18
References
22
Work Report
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Viscous Incompressible Flow in Over a Step 2009
Problem Formulation
Problem statement
Incompressible flow was considered inside a 2-D back step configuration. The Backstep geometry is
shown below as it was set up in the experiment.
Figure 1: Three Dimensional Duct Geometry
The Heights of the step are H = 0.51485 and
h = 0.48515
Figure 2: 2D Backstep Channel Geometry with lengths to be measured
The above figure shows the configuration of a backward facing step which is referred to here as the
backstep channel. The origin of the physical plane coordinates is placed at the location of the step
transition.
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Viscous Incompressible Flow in Over a Step 2009
U in 
Ur
.
H
The
The mean velocity Uin at the inlet is related to the outflow velocity by the relation
U r Lr
Re 
 , the subscript ‘r’ refers to reference quantities at the outlet.
Reynolds number is defined as
For the above given backstep channel geometry, the region extending from slightly upstream of the
backward facing step to the furthest reattachment point is referred to as the “transition region”. In this
region, convection dominates over diffusion. Using Gambit we were asked to create a structured grid
distribution for laminar viscous flow through this backstep channel at two different Reynolds Numbers:
Re = 300 and Re = 507. The Graphical representation of the grid selected for these configurations is also
provided. Streamline and vorticity contours inside the channel along with the the total velocity profiles at
various x-locations inside the channel were to be plotted and analyzed. Lengths L1, L4 and L5 (seen in
figure 2) for these cases were computed and compared with the experimental data of Armaly et al. (1980)
that was provided to us.
Theory
This project focuses on the recirculation zone created by a backwards facing step. A recirculation zone is
created when there is an unfavorable pressure gradient in the flow path. If the pressure gradient increases
too much, the flow will separate way from surface it was following. Once this separation occurs, the
pressure will decrease in the area between the separated flow and the surface. This will cause the fluid to
reverse the flow direction and recirculate. The flow will only reattach if the pressure gradient is sufficient
enough for it. As a result, higher Reynolds numbers (higher velocity) produce a predicted longer length
before reattachment. At Reynolds numbers higher than those tested in this report, additional recirculation
zones are seen to form.
For this project, the flow will pass over the step and separate causing a recirculation zone. The pressure
gradient will decrease enough for the flow to reattach to the bottom surface of the channel. But this will
occur too quickly for the fluid passing of the upper surface. The pressure gradient on the upper surface
will increase too far causing the upper surface to separate and recirculate. The flow will eventually
reattach further downstream in the channel.
Figure 3: Theoretical and experimental values for velocity profiles for Re = 389 (Armaly)
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Viscous Incompressible Flow in Over a Step 2009
Recirculation and separation can be seen elsewhere other than a backwards facing step. Take an airfoil
for instance. When the angle of attack for an airfoil becomes too great, the flow on the suction side at the
trailing section begins to separate off and recirculate back. This is what causes an airfoil to stall. In that
case, a recirculation zone has a negative impact and is not preferred. However, in the combustion
chamber of a gas turbine engine, the flame is only sustained because of the recirculation zone created by
the flame holder. In fact, the combustion chamber design of a scramjet closely resembles that of this
project. The recirculation zone at the step is the flame holder of a scramjet.
Gambit and FLUENT software
Gambit
It was determined that, in order to get accurate data corresponding to the cases tested, the flow would
need to be fully developed before the step. To create a fully developed flow, the inlet length of the
channel would have to be greater than or equal to the hydraulic entrance length. The entrance length for
each velocity was determined by the formula used in project 1. l/Dh = 0.05Re, where Dh is the hydraulic
diameter (4A/P, where A is the cross-sectional area and P is the perimeter of the wall). For an infinite
channel, the hydraulic diameter is 2 times the height. For the Re = 300 case, the entrance length of the
inlet was determined to be 15.01 m. To be sure fully developed flow, a length of 18 meters was meshed
for the inlet. The expanded region was 20, as experimental data showed it should reach steady state by
that length. For the Re = 507 case, the entrance length was found to be 25.37 m, so a total length of the
inlet was taken to be 32 m. The length of the expanded region was taken to be 25 to accommodate the
higher length before coming to steady state for the higher Reynolds number, as found by experimental
data.
Because the same number of points were distributed across the upper and lower left edges of the grid,
which have different lengths, a different spacing occurred. This different spacing, when the full volume
was meshed, resulted in skewness, as the even spacing on the right boundary was adjusted to. To resolve
this problem and reduce skew, two virtual lines were created, breaking the mesh into 3 rectangles. Each
were then meshed separately. However, this mesh did not fully converge, reducing to a 10-2 residual, and
remaining stably at that value. To resolve the problem, a clustering of 1.005 successive ratio towards the
walls (except on the bottom of the entrance) was used to better resolve the boundary layer.
For the Re = 300 case, 100 nodes were taken at the outlet, and 50 each were distributed on the left
corresponding sides (the vertical wall and the inlet). 175 nodes each were taken for the top and lower
horizontal wall sections for the inlet and expanded region. The Re = 300 mesh is shown in figure **.
Figure 4: Mesh for Re = 300 case
For the Re = 507 case, there are 160 nodes along the outlet, 80 along the inlet, and 80 along the left
vertical wall. 200 nodes along each bottom wall, and 400 along the top wall were used, with each block
being meshed separately to avoid skew. The Re = 507 mesh is shown in figure **.
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Viscous Incompressible Flow in Over a Step 2009
Figure 5: Mesh for Re = 507 case
FLUENT
In FLUENT, the pressure based, implicit, two dimensional, Green- Gauss cell based solver was chosen.
Steady and unsteady were compared for results against each other. A second order upwind discretization
was chosen. Operating conditions were set to the default 1 atmosphere of pressure. The inlet was a
velocity inlet set to the average initial velocity desired.
For viscosity, the laminar case was chosen as the Reynolds numbers, by definition of the problem
statement are low.
Results:
Because in some cases a separated flow such as in the step case can become unstable, both a stable and
unstable case were run for each Reynolds number, for comparison. In each case, the values for the length
of separation and reattachment for the upper and lower walls is seen to correspond closely to the
experimental data from Armaly’s results presented in Ghia’s paper on simulation of this case.
300 steady
300 unsteady
Fluent
Reference % error
Fluent
Reference
% error
L1 (bottom)
4.96641
4.96
0.129
4.96641
4.96
0.129
L4 (top1)
4.10146
4.05
1.271
3.99574
4.05
-1.340
L5 (top2)
7.16442
7.32
-2.125
7.28597
7.32
-0.465
507 steady
507 unsteady
Fluent
Reference % error
Fluent
Reference
% error
L1 (bottom)
6.48443
6.08
6.652
6.22821
6.08
2.438
L4 (top1)
5.10633
4.7
8.645
4.86379
4.7
3.485
L5 (top2)
11.8713
11.45
3.679
12.0273
11.45
5.042
Figure 6: Numerical comparison of L1, L4, and L5 to for the same Reynolds number (Taken from Ghia, 1983)
As one can see from figure 6, cases were generally well matched, within 10% for all numbers, and within
less for the lower Reynolds number case. While the unsteady and steady cases gave slightly different
results, neither is seen to be overall better or worse at prediction of all characteristics of the flow. As
predicted by theory, the lengths of separation and reattachment are lengthened for the higher Re of 507.
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Viscous Incompressible Flow in Over a Step 2009
Case1: Re = 300, steady case
To most accurately determine the distance along the flow of the separation and reattachment points
corresponding to lengths 1, 2, and 3 in the paper, the shear stress was taken along the wall. When the
flow separates or reattaches, the opposite flow directions on either side lead to zero tangential velocity at
the wall. Because shear stress is proportional to the strain rate (du/dy) and the velocity u is zero , the
shear stress is zero at the separation and reattachment points. The shear stress data correspond well with
theory, clearly showing 3 distinct points where shear decreases to zero and increases again. In figure 7,
one can see the initial zero shear stress due to the wall increasing, then coming to zero at L1 = 4.966 and
returning. Figure 7 shows the initial positive shear stress going to zero at L4 first, increasing in the
recirculation zone, then returning to zero at L5, then rising for the rest of the channel.
Figure 7: Wall shear stress for lower wall, Re = 300 steady case
Figure 8:: Wall shear stress for upper wall, Re = 300 steady case
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Viscous Incompressible Flow in Over a Step 2009
Figure 9: stream lines for Re = 300, steady case
In figure 9 one can see the stream function contours. The dark regions correspond to where the stream
lines (which run along lines of constant stream function) have separated from the wall, and correspond
well with the experimental and simulation data with respect to the separation and reattachment points. In
figures 10 to 12 below, one can see the development of the velocity profile along the channel and step. At
the inlet, a flat constant velocity profile is seen. This develops along the next 2 profiles to fully developed
parabolic channel flow. In figure 11, a close-up of the initial separation of flow from the lower wall is
seen and backflow in the lower part of the recirculation zone is observed by the reversal of direction in
the velocity vectors. Figure 12 then shows the separation from the upper wall and reattachment of the
flow.
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Viscous Incompressible Flow in Over a Step 2009
Figure 10: Early velocity profiles along channel Re = 300, steady case
Figure 11: Closeup of velocity profiles at step, showing separation, Re = 300, steady case
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Viscous Incompressible Flow in Over a Step 2009
Figure 12: Closeup of later velocity profiles showing upper wall separation and reattachment, Re = 300, steady case
Figure 13: Vorticity contours, Re = 300, steady case
Figure 13 shows the vorticity contours. One can clearly see the high vorticity region attached at the wall
separate and become a free shear layer first at the step, then at the upper recirculation zone boundary. It
then reattaches and settles to a lower value (due to the lower velocity in the wider portion of the channel).
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Viscous Incompressible Flow in Over a Step 2009
Case 2: Re = 300, unsteady case
Below one can see the wall shear stress plots for the lower and upper wall. Compared with the steady
case, the shapes are nearly identical and the reattachment points within a few percent.
Figure 14: Wall shear stress for lower wall, Re = 300 unsteady case
Figure 15: Wall shear stress for upper wall, Re = 300 unsteady case
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Viscous Incompressible Flow in Over a Step 2009
Figure 16: Residual plot, Re = 300 unsteady case
Figure 17: Streamlines for Re = 300 unsteady case
In figure 16, the residual plot showing convergence of the time dependent solution occurring. Figure 17
shows the streamlines, again clearly separating around the recirculation zones, with little of the flow
entering or leaving them. Figures 18 and 19 illustrate the velocity profiles as they follow the back flow
region below and above, as the velocity profile develops and reattaches.
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Viscous Incompressible Flow in Over a Step 2009
Figure 18: Velocity profiles for Re = 300 unsteady case
Figure 19: Closeup of velocity profiles, Re = 300 unsteady case
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Viscous Incompressible Flow in Over a Step 2009
Figure 20: Vorticity contours for Re = 300 unsteady case
The vorticity contours for the unsteady case of Re = 300 look very much like the vorticity contours of
case 1, with the local maximum line of vorticity following the free shear layers around the recirculation
zones.
Case 3: Re = 507, steady case
The 507 case converged relatively directly, but at a high number of iterations, as seen in figure 21. As
can be seen in 22 and 23, the lengths are increased due to the higher velocity requiring a longer length for
the same expansion at a critical pressure ratio.
Figure 21: Residual plot for Re = 500 steady case
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Viscous Incompressible Flow in Over a Step 2009
Figure 22: Wall shear stress for upper wall, Re = 507 steady case
Figure 23: Wall shear stress for lower wall, Re = 507 steady case
Figures 23 and 24 show the wall shear stress for the upper and lower wall respectively. Again one can
see the lower wall with initially zero shear stress (as it is in the corner, where u = 0 by definition),
increases in magnitude, and then decreases to zero again at 6.484. Likewise one can again see the upper
wall with an initially high shear stress followed by the two zero points corresponding to lengths L4 and
L5, then settling to a lower value downstream.
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Viscous Incompressible Flow in Over a Step 2009
Figure 24: Streamlines for Re = 507 steady case
The streamlines above show the same characteristics as with both Re = 300 cases, though with the lengths
increased and larger recirculation zone volumes. This is further clarified by looking at the velocity
profiles below in figures 25 and 26. Note again the negative velocity toward the bottom of the close-up of
the first recirculation zone at the step.
Figure 25: Velocity profiles for Re = 507 steady case
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Viscous Incompressible Flow in Over a Step 2009
Figure 26: Closeup of velocity profiles for Re = 507 steady case
Figure 27: Vorticity contours for Re = 507 steady case
Figure 27 shows the vorticity contours for the Re = 507 case. As in the first two cases, one can see the
free shear layer outlining the two recirculation zones. However, both the length of the recirculation zones
and the magnitude of the vorticity are seen to be significantly increased due to the higher velocity.
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Viscous Incompressible Flow in Over a Step 2009
Case 4: Re = 507, unsteady case
Of all the cases, the higher Reynolds number (507) unsteady case took the most iterations to converge, as
can be seen in figure 28 of the residual history. The unsteady case in other respects gave the same order
of accuracy in the solution, falling within a few percent of the expected value for lengths.
Figure 28: Residual plot for Re = 507 unsteady case
The unsteady case Re = 507 wall shear stress plots are seen to be nearly identical to the plots for the
steady case, as it was run until a steady state was reached.
Figure 29: Wall shear stress for lower wall, Re = 507 unsteady case
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Viscous Incompressible Flow in Over a Step 2009
Figure 30: Wall shear stress for upper wall, Re = 507 unsteady case
Figure 31: Streamlines for Re = 507 unsteady case
The streamlines, as seen in figure 31, correspond to the same path of the flow seen in experiment and the
steady case for Re = 507. The velocity profiles below again show the structure of the recirculation zone
with the back flow at the wall in each separated zone.
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Viscous Incompressible Flow in Over a Step 2009
Figure 32: Velocity profiles showing both separations and reattachments Re = 507 unsteady
Figure 33: Closeup of First recirculation zone velocity contours Re = 507 unsteady case
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Viscous Incompressible Flow in Over a Step 2009
Figure 34: Vorticity contours for Re = 507 unsteady case
Above are the vorticity contours for the Re = 507 unsteady case. It is nearly identical in shape and
dimension to the steady case, with the same higher magnitude vorticity outlining the boundaries of the
two recirculation zones before coming to a steady case further down the channel.
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Viscous Incompressible Flow in Over a Step 2009
References
Ghia, K.N.; Osswald, G. A.; Ghia, U. “A Direct Method for the Solution of Unsteady TwoDimensional Incompressible Navier Stokes equations” Proceedings of Second
Symposium on Numerical and Physical Aspects of Aerodynamic Flows, Editor: T.
Cebeci, California State University, Long Beach, CA, Jan 17-20, 1983.
Armaly, B.F., Durst, F., Pereira, J. “Experimental and Theoretical Investigation of Backward
Facing Step” Journal of Fluid Mechanics, vol. 127 pp. 473-496, 1983
Anderson, John D. Jr., Computational Fluid Dynamics The Basics With Applications. McGrawHill, Inc. New York, 1995
“Near-Wall Mesh Guidelines for Wall Functions” webpage, Fluent inc. 2007
http://web.njit.edu/topics/Prog_Lang_Docs/html/FLUENT/fluent/fluent5/ug/html/node37
3.htm
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Viscous Incompressible Flow in Over a Step 2009
Work Report
Shiva Naraharisetty – Worked on steady cases in FLUENT, cowrote report.
Robbie Driscoll – Worked on preliminary cases in FLUENT, cowrote report.
Sandeep Kumar – Generated meshes in Gambit. Worked on unsteady FLUENT case, cowrote
report.
William Stoddard – Performed analysis/post processing. Cowrote report.
Rajiv Kattekola – Worked on unsteady FLUENT case. Cowrote report.
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