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Alex Wright
100963406
1
MECH 5001: THEORY OF VISCOUS FLOW
ASSIGNMENT 1
Alex Wright
100963406
ABSTRACT
A cross flow over a cylinder in a quasi-3-dimensional domain
was simulated using FINE/Hexa. A mesh independence study
was conducted to discover a mesh with greater than 300,000
cells was necessary to accurately resolve physical flow
structures. The simulation convergence was adversely
affected
by the use of a steady state, time invariant solver for a
naturally
dynamic physical flow. The drag force was used to conduct
the
mesh independence study. The drag coefficient was determined
to be approximately 0.8 based on the simulation. This is in
contrast to the empirically determined value of approximately
1
for this flow with Reynolds number on the order of 103. A
combination of factors can be attributed to the discrepancy
between observed and expected results. They include the
domain
geometry and the computational strategy. This exercise can
be
considered a success despite the poor quality of results
obtained
via numerical simulation.
INTRODUCTION
The purpose of this assignment is to introduce the students
to the practice of computational fluid dynamics by
investigating
a cross flow over an infinite cylinder. This is a simulation
that
has been investigated in great detail experimentally and
numerically. This experiment is popular for its geometric
simplicity while providing a variety of flow structures in
the
wake of the cylinder at various flow conditions.
Experimental investigations have developed an empirical
relationship between the Reynolds number and the drag
coefficient of the flow. The relationship is shown in Figure
1.
Figure 1: Reynolds number versus Drag Coefficient for an
infinite
cylinder. (Weisstein)
Depending on the Reynolds number, the relationship
between the pressure driven drag forces and the total
observed
drag force can change. For very low Reynolds numbers the
drag
is dominated by skin friction and 1
. At moderate
Reynolds numbers, the total drag is roughly equal to the
pressure
driven drag and = 1. At very high Reynolds numbers, the flow
becomes turbulent modifying the fluid properties in the
boundary layer. At this point, there is a brief drop as
separation
is delayed. The drag coefficient then continues to increase
with
Reynolds number.
This study will use the drag force as a method of mesh
independence validation. The observed drag will be compared
to
the expected value determined by the equation
= 1
22
Where , the drag coefficient, is determined from Figure 1, is
the fluid density, is the free stream velocity and is the
interfacial area perpendicular to the flow.
METHODS
The domain for this study was modeled in Pro|Engineer
Wildfire 4.0, a solid modeling package. The domain is shown
in
fig 2, which includes the dimensions
Figure 2: The domain geometry. DCylinder=10mm (Matida)
The solid model was triangulated and imported into the
domain meshing software Hexpress 2.10-4. This software
develops fully hexahedral unstructured meshes for complex
geometries. The software allows for mesh refinement around
desired geometry features as well as automatic insertion of
a
viscous boundary layer of cells to ensure a non-dimensional
wall
distance of y+=1.0 for accurate modeling of the viscous
layer.
The flow solver used in this study was FINE/Hexa 2.10-4.
It is an unstructured, density based, finite volume solver
which
solves the Reynolds Averaged Navier Stokes equations.
Convective fluxes were discretized via Roes second order upwind
scheme. Diffusive fluxes were discretized via the central
difference scheme. The general Navier-Stokes equation solved
by FINE/Hexa is
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Alex Wright
100963406
2 Copyright 2014 by The Crown in Right of Canada
+
S
S
=
Where is the control volume, is the control surface, is
the set of conservative variables, is the advective fluxes, is
the diffusive fluxes and contains the source terms. These are
further defined in the Theory Manual for FINE/Hexa.
(Numeca International) The relevant chapter is included in
Appendix A for convenience.
The boundary conditions are outlined in Figure 4. The inlet
velocity was 6 m/s, orthogonal to the inlet plane. The
outlet
pressure was set as 101300 Pa, or 0 Pa gauge. The domain
walls
were set as inviscid in order to approximate a 2-dimensional
flow
using this quasi 3-dimensional domain. The cylinder wall was
a
standard wall with a no-slip condition applied. These
conditions
yield a Reynolds number of approximately 4x103, where 1
according to Figure 1.
Figure 4: Domain Boundary Conditions. Inlet - Green, Outlet -
Red,
Domain Walls - Blue, Cylinder Wall - Black.
MESH INDEPENDENCE
This study was initiated with a very coarse mesh of only
8256 cells. The mesh was successively refined in 5 steps up to
a
maximum of 518896 cells. The meshes are shown in Figure 3.
The intermediate meshes are listed in Table 1. Table 1 relates
the
number of cells to the drag force on the cylinder calculated
by
the flow solver. The empirical expectation of the drag force
from
= 1
22 is 0.000425 N.
Table 1: Mesh Independence Data
Simulation Cells FD (x10-4)N
1 8256 4.61
2 31764 3.14
3 127808 4.60
4 326696 3.23
5 518896 3.78
6 264299 3.26
The poor convergence of the drag force can be attributed to
the unsteady nature of the simulation. The coarse meshes
converged well because they could not properly resolve the
dynamic vortex shedding in the wake of the cylinder. As the
cell
count passed 100,000, the mesh became fine enough to resolve
the vortices. The vortices are physical and are known as von
Karman vortices. The vortex shedding made steady state
convergence difficult to attain. The drag force calculated by
the
solver would oscillate as the vortices developed. While the
mesh
independence criterion did not converge to the expected
value,
they did converge to a value near 0.00032 N. Therefore, one
may
assume that the simulations are mesh independent around
300,000 cells and could capture physical phenomena. The
deviation from the expected value for drag force may have
been
due to the choice to use a quasi-3-dimensional model rather
than
a truly 3-dimensional or 2-dimensional model for an infinite
cylinder. The numerically calculated drag force yield a drag
coefficient of around 0.8.
Figure 3: Computational Meshes. From top to bottom: Coarsest
Mesh,
Finest Mesh, and Finest Mesh near Cylinder
-
Alex Wright
100963406
3 Copyright 2014 by The Crown in Right of Canada
DISCUSSIONS AND RESULTS
The following figures are from simulation 5, which had
approximately 500,000 cells.
Figure 5: Velocity Vector Plots. Top: Whole Domain. Bottom:
Zoomed
in near Cylinder Wake
Figure 5 shows the velocity vector plot at the central x-y
plane of the domain. The unsteadiness of the wake can be
clearly
observed in this plot.
Figure 6: Magnitude of Velocity with Velocity Streamlines
Figure 6 shows the magnitude of velocity contour plot with
velocity vector streamlines. This plot clarifies the instability
in
the wake of the cylinder. The streamlines indicate the
direction
of flow and can clearly be observed forming discrete vortex
cores. The streamlines illustrate the complexity of the flow in
the
wake as well as the vortex cores developing therein.
Figure 7: Total Pressure Contour Plot
The total pressure contour plot in Figure 7 displays the low
pressure zone within the wake of the cylinder. The lower
total
pressure indicates a loss of energy in the flow as it passed
the
cylinder. The loss is a combination of skin friction and
viscous
mixing in the wake.
Figure 8: Static Pressure Contour Plot and with Velocity
Streamlines
The static pressure contour plot in Figure 8 shows regions
of the wake with lower static pressure. Fluid is drawn to
regions
of low static pressure from regions of higher static
pressure.
These low pressure regions are clearly indicated by the
velocity
streamlines as vortex cores. As this simulation approaches a
dynamics steady state, the vortex shedding would become more
regular in size and location. These simulations had not
reached
the dynamic steady state.
-
Alex Wright
100963406
4 Copyright 2014 by The Crown in Right of Canada
CONCLUSIONS
The simulation of cross flow over a cylinder using a quasi-
3-dimensional domain was an interesting exercise with a
myriad
of challenges and fascinating results. The dynamic nature of
the
flow did not lend itself to a simple simulation. The
simulation
had a Reynolds number around 4x103 which is in the laminar
flow regime wherein vortices are periodically shed in the
wake
of the cylinder. The vortex shedding resulted in fluctuations
of
convergence criteria for the steady state solver. Regardless of
the
convergence difficulties, the dynamic flow structures were
resolved when the mesh was fine enough (n>300,000). The
vortex cores have been shown to be regions of low static
pressure
separating from the aft end of the cylinder. In regard to the
drag
force and the discrepancy observed between the computed and
expected value could be due to the geometry of the domain as
well as the dynamic nature of the flow. A
quasi-3-dimensional
domain does not allow for true 3- or 2-dimensional
observation
of the phenomenon. The cylinder therefore was not truly
representative of an infinitely long cylinder. A better strategy
for
simulating this flow would be to use mirror or periodic
boundary
conditions on the domain side walls and use an unsteady flow
solver to capture the dynamic vortex shedding.
REFERENCES
Matida, Edgar. "Assignment 1." Ottawa, January 2014. Numeca
International. "Theoretical Manual FINE/Hexa."
Brussels: Numeca International, February 2010.
Weisstein, Eric W. Cylinder Drag. 4 February 2014.
.