Unsteady Analysis of Separated Aerodynamic Flows using an
Unstructured Multigrid Algorithm
Old Dominion University
NASA Langley Research Center, Hampton, Virginia
Old Dominion University
An implicit method for the computation of
presented. The resulting nonlinear system of
equations is solved at each time step using an
agglomeration multigrid procedure. The method
allows for arbitrarily large time steps and is
efficient in terms of computational effort and
storage. Validation of the code using a one-
equation turbulence model is performed for the
well-known case of flow over a cylinder. A
Detached Eddy Simulation model  is also
implemented and its performance compared to
the one equation Spalart-Allmaras Reynolds-
model . Validation cases using DES and
RANS include flow over a sphere and flow over
a NACA 0012 wing including massive stall
regimes. The project was driven by the ultimate
aerodynamic interest, such as massive stall or
flows over complex non-streamlined geometries.
The use of time-dependent simulations for flows
of practical interest is much less widespread than
the use of steady-state flow simulations, due to
limitations related to computational time and
computational resources. However, many flows
are inherently unsteady, particularly when large
amounts of separation are present, and must be
simulated as such in order to obtain meaningful
Copyright c 2001 by AIAA, Inc. All Rights Reserved.
Transient flow simulations can either be based
on explicit or implicit time-stepping schemes.
allowable time step size of an explicit scheme,
which is proportional to the smallest cell size in
the entire computational mesh. Explicit schemes
are well suited for unsteady applications in
which the time scale of interest is comparable to
the spatial scales. In these cases the mesh should
be clustered only in regions of interest where it is
absolutely necessary or the explicit time step can
become unnecessarily small.
Explicit time steps may become too restrictive
for cases characterized by larger time scales or in
cases where there is a wide variation in the grid
resolution. In these cases it is desirable to
develop a fully implicit method in which the
time step is only determined by the physics of
the flow and not by the cell size. At each time
step the unsteady residual must be driven to zero
and this is usually done using inner iterations.
The number of inner iterations needed for each
time step is related to the problem being solved
and the size of the physical time step used.
To develop an unsteady solver that can provide
an accurate description in time of the flow is the
first requirement to address before attempting to
solve flows with massively separated regions
that are inherently unsteady. On the other hand,
it is also important to review the different
techniques available to compute turbulence,
which is at the core of separated flows, and will
be a key parameter to determine how well the
numerical scheme captures all the physical
aspects of the flow. In this sense, numerical
solutions of turbulent flow cases can be achieved
using different levels of approximation. The
(RANS). In the RANS equations, the turbulent
fluctuations appear in the Reynolds stress term
that must be modeled using any of the turbulence
models available in the literature. However, a
common limitation of these models is their lack
of generality since the model coefficients are
usually set using simple well-documented flows.
In this sense, current RANS solvers are fairly
successful at predicting mostly attached flows,
such as a wing in cruise condition, but fail to
capture a range of different off-design situations
as post-stall regimes,
configurations, and non-streamlined bodies. In
general, in the cases in which the RANS
approach fails, the flow is characterized by large
amounts of separation in which a very wide
range of scales is present in the flow. While the
small scales tend to be universal in nature, the
larger scales are affected by the boundary
conditions. This is the main cause of the lack of
generality of turbulence models, as it is difficult
to model the effect of the large scales in the same
way for many different types of flows.
The failure to develop a universally valid
turbulence model has led to alternate approaches
such as Direct Numerical Simulation (DNS) and
Large Eddy Simulation (LES). DNS is the most
straightforward approach to the problem. DNS
consists in solving the governing equations on a
mesh fine enough to capture the smallest scales
contained in the flow with a scheme designed to
dissipation. The drawback is extremely high cost
of the DNS computation, which is proportional
to at least Re3. For these reasons, DNS is
generally limited to very simple flows and low
The flow limitations of RANS and the difficulty
of using DNS for realistic applied engineering
problems have generated a great interest in the
Large Eddy Simulation approach (LES) for
separation. Large Eddy Simulation is a technique
Reynolds Averaged Navier-Stokes . In LES
the contribution of the large scales is computed
exactly and only the smallest scales in the flow
are modeled. However, a major difficulty of
flows, is that near solid surfaces all the eddies
are small and the “large” and “small” eddies tend
to overlap. Therefore the required grid spacing
and time step gradually fall towards DNS as the
solid boundary is approached [4 ].
Using LES to resolve near wall streaks would
bring an immense penalty at industrial Reynolds
numbers. In this line of reasoning the Detached
Eddy Simulation (DES) approach was conceived
with the idea of combining the strengths of
Reynolds Averaged methods near the solid
boundaries and of Large Eddy Simulation
The ultimate goal of this work is to develop a
large eddy simulation capability based on an
existing unstructured grid Navier-Stokes solver
 to be able to perform detached eddy
simulations combing RANS near the walls and
LES in massive separated regions in a non zonal
manner and to compare the solutions obtained
using this approach to the solutions obtained
using a classical RANS approach with a Spalart-
Allmaras one equation turbulence model .
2.Unsteady Reynolds Averaged Navier-
2.1 Steady solver description
The Reynolds averaged Navier-Stokes equations
are discretized by a finite volume technique on
meshes of mixed elements, including tetrahedra,
pyramids, prisms and hexahedra. In general,
prismatic elements are used in the boundary
Flow variables are stored at the grid vertices and
a single unifying edge-based data-structure is
used to handle all elements of the grid.
Convective and viscous fluxes are discretized
along edges using a central difference finite
The non-dimensional steady conservative form
of the full Navier-Stokes equations can be
where w represents the solution vector of
conserved variables and R represents the spatial
vanishes at the steady state.
or residual, which
An implicit solution procedure for solving
equation (1) begins with the linearization of the
residual about the current time step or iteration
which can be solved as:
where ÿ is a pseudo-time used to advance the
solution, and ∆w represents the new correction to
be applied to the solution. Rather than inverting
the large Jacobian in equation (3), a reduced
Jacobian which is simpler to invert is employed.
In regions of isotropic grid cells, only the block
diagonals of the Jacobian are retained, leading to
a block-Jacobi or point-implicit scheme. In the
stretching is present, lines are constructed in the
direction normal to the solid wall boundary and
the Jacobians along these lines are inverted,
using a block tridiagonal solution algorithm.
This procedure relieves the stiffness associated
with high grid stretching in these regions, thus
providing more rapid convergence [7 ].
This locally implicit scheme is used as the
smoother on all levels of an agglomeration
multigrid algorithm.The multigrid algorithm
constructs coarse level grids automatically by
fusing together neighboring fine grid control
volumes to form a smaller number of large
coarse grid control volumes.
acceleration is achieved by cycling back and
forth between the fine and coarse grid levels of
the multigrid sequence, using the locally implicit
solution procedure as a solver on each grid level
The unstructured multigrid solver is parallelized
by partitioning the domain using a standard
between the various grid partitions running on
individual processors using the MPI message-
passing library . The solver can be run on
including clusters of personal computers.
2.2 Unsteady term implementation
The unsteady form of the governing equations is
obtained by adding the time derivative to
where R(w) denotes the discretization of the
spatial derivative terms, as previously.
Making use of a second-order accurate three
point backward approximation for the time
derivative and evaluating R(w) at time level
(n+1), we obtain [10,11,12]:
We may now define and unsteady residual as:
where wn+1is the approximation to w and the
source term S(wn,wn-1) remains fixed throughout
the solution procedure at each time step. In this
form, the non-linear problem to be solved at each
time-step reduces to R*(w) = 0.
redefinition of the residual, the same iterative
multigrid procedure employed to solve the
steady-state problem can be used to solve the
non-linear unsteady residual at each physical
time step [10,11,12].
The solution procedure consists of an outer loop
over the physical time steps, used to advance the
problem in time, and an inner (multigrid) loop
over pseudo-time used to drive the unsteady
residual to zero at each time level. While the
maximum size of the physical time step is only
determined by the physics of the problem, the
convergence of the inner sub-iterations for each
time step varies inversely with the size of the
physical time step.
3.Unsteady solver validation
The flow around a circular cylinder is a well-
known case, which has been widely studied
computationally and experimentally. This case is
used as the basis for validation of the unsteady
RANS solver, and for assessing grid resolution
observed in the cylinder flow. Two different
meshes of 252,000 and 631,000 grid points and
three different time steps of 0.5, 0.25 and 0.1
were used. The time is non-dimensionalized as t
= to/(d/U∞) where d is diameter of the cylinder
and U∞is the freestream velocity.
Figure 1: 3D View of cylinder and lateral walls
The one equation Spalart-Allmaras turbulence
model  was used for all calculations in fully
turbulent mode. In all cases the agglomeration
multigrid strategy was used with four levels. The
Mach number is 0.2 and the Reynolds number is
1200 for this case. All runs were performed in
parallel using 16 processors of a Pentium PC
cluster at ICASE.
The computational domain in the plane normal
to the cylinder span has an aspect ratio of 1 and a
side length of 100 cylinder diameters. A span of
two cylinder diameters is employed, and inviscid
(slip velocity) boundary conditions are applied at
simulations reported herein were also compared
dimensional unstructured solver , and found
to agree well in terms of force coefficient
histories and shedding frequency.
Table 1 shows the Strouhal Numbers computed
for each mesh and each time step of the three-
achieved as the time step is reduced and the
mesh size increased. A second-order accuracte
convergence behavior is observed as the time-
step is reduced, validating the accuracy of the
three-point backwards difference scheme used to
discretize the time step. From the smallest time
step results, the solution can be seen to be grid
converged, at least with respect to the prediction
of the vortex shedding frequency. The computed
Strouhal number compares very well to the
experimental value of St = 0.21 given by Dresher
[13,14]. Figure 2 shows the time history of the
lift coefficient, while the oscillatory pattern
corresponding to the vortex shedding is shown in
Table 1. Predicted Strouhal Number for Various
Grid and Time Step Sizes
Figure 2: Mach contours at two different time
states for flow over circular cylinder. Mach =
0.2, Re = 1200.
Figure 3. Computed LiftCoefficient Time History
for Flow over Circular Cylinder using three
different time steps. Mach = 0.2, Re = 1200
Figure 4. Mean Surface Pressure Coefficient
Distribution for Flow over Circular Cylinder at
Mach = 0.2, Re = 1200, Compared with
Experimental Data at Various Reynolds
Figure 4 shows the pressure distribution over the
surface of the cylinder. The pressure distribution
was computed by averaging results at different
times distributed along several oscillations. For
this calculation the finest mesh and the smallest
time step was used. As can be observed from
Figure 4, the computed pressure distribution
compares closely to experimental results at a
higher Reynolds number than the one used for
these computations. This is likely due to the use
of the turbulence model in fully turbulent mode,
in order to avoid the issues of transition
prediction. Similarly, the backpressure obtained
of Cpb= -1.20 compares closely to the Cpb= -
1.20 at Re=27,700 measured by Linke [13,15].
The mean value of the drag coefficient computed
is Cd=1.3 compared to Cd=1.2 as measured by
Wieselsberger [13,16] for Re=30,000.
4. Detached Eddy Simulation.
Detached Eddy Simulation (DES) is a hybrid
technique that combines RANS and LES in a
non-zonal manner. DES is based on the Spalart-
Allmaras one equation RANS model  in
which the length scale d, which is traditionally
taken as the nearest distance at any given point to
the closest wall, is replaced as the minimum
between the distance to the wall and a length
proportional to the local grid spacing:
dDES= min (d, CDES∆x)
where CDESrepresents a model constant which
has been taken as 0.65 in previous work [1,17].
Traditionally, on structured grids [1,17,18], ∆x is
taken as the maximum grid spacing over all three
directions. In our particular case, the definition
of ∆x has been modified for unstructured grids
by taking it as the maximum edge length
touching a given vertex. In boundary layer
regions, ∆x far exceeds the distance to the wall
d, and the standard Spalart-Allmaras RANS
turbulence model is recovered. However, away
from the boundaries d exceeds CDES∆x and the
models turns into a simple one equation sub-
grid-scale (SGS) model with the mixing length
proportional to the grid spacing. This effect is
illustrated by plotting contours of the distance or
length scale function for both the RANS and
DES models in Figure 5, where it is observed
that both models employ the same length scales
near the wall, but use vastly differing length
scales in the regions far removed from the wall,
where the DES model reverts to an LES mode
and a Smagorisnky–like expression for the eddy
viscosity is obtained.
Figure 5. Distance function/Length Scale
comparison between URANS and DES.
5.1 Flow around a Sphere.
DES is applied to predict the flow around a
sphere. Similarly to the flow around a circular
cylinder, this case has been widely studied using
experimental and computational approaches. It
has been shown that over a wide range of
Reynolds numbers (280< Re < 3.7x105) the flow
is characterized by the vortex shedding with
large-scale vorticity emanating from the shear
layer which separates from the surface of the
sphere. In our case a Reynolds number of 104is
used, corresponding to the sub-critical regime
(laminar boundary layer separation). This is
comparative basis for our results.
The main goal of this case is to compare the
results obtained using traditional URANS and
DES. The comparison includes the time history
of integral parameters such as drag coefficient
and mean distribution of pressure around the
5.2.1 Numerical approach.
The flow over a sphere is calculated for a
Reynolds number of 104and a Mach number of
0.2 on an unstructured mesh of 767,000 vertices.
The computational domain is a cube of 100
sphere diameter lengths in each direction. This
large domain is chosen to ensure that the
downstream boundary condition does not lead to
spurious oscillations within the domain. Near the
surface of the sphere, the normal grid spacing is
10-4× d, where d represents the diameter of the
For the unsteady calculations two different time
steps of 0.1 × d/U∞and 0.05 × d/U∞were used.
In all cases the multigrid strategy was used with
four grid levels. All runs were performed on 32
processors of a Pentium PC cluster at ICASE.
For the DES runs, the CDESvalue used was 0.65.
This value was chosen based on previous studies
by Shur et al. , where the value was calibrated
in isotropic turbulence.
Constantinescu et al. 
calculations of the flow over a sphere at Re =
104. A parallel study is being conducted to
examine the effects of variations in the value of
CDES using the same test case of decaying
homogeneous turbulence in a box as done by
Shur et al. .
prove this value of
5.2.2 Code validation.
Following the procedure used by Constantinescu
et al. , the accuracy of the numerical
approach was established by comparing the
results obtained in the steady regime at Reynolds
number 250 with previous computational and
The drag coefficient is compared in Table 2 to
other simulation results and experimental data.
Because previousresultswerebased on
incompressible simulations, and the current
formulation, the importance of compressible
effects was also investigated by running the
simulation at Mach numbers of 0.2 and 0.1, both
satisfactory for all the cases tested.
The agreement is
Johnson and Patel
0.70 - 0.72
M = 0.1
M = 0.2
M = 0.1
M = 0.2
Table 2. Computed Steady Drag Coefficient for
Flow over Sphere at Re = 250 compared with
Experimental and Previous Computational
5.2.3 Unsteady Results.
For the unsteady runs, the flow around a sphere
is computed at a Mach number of 0.2, without
preconditioning, and a Reynolds number of 104.
At this Reynolds number the detached vortex
sheet from the sphere is fully turbulent while the
boundary layer on the sphere remains laminar.
The Strouhal number associated with the vortex
shedding at this Reynolds number is in the range
of 0.185-0.200 depending on the investigation.
The large scatter of the data is mainly due to the
methods of the different investigations.
Figure 6. DES Time history of Drag Coefficient
for flow over Sphere at Mach = 0.2, Re = 10,000
using two different time steps
Figure 7. URANS Time history of the drag
coefficient for flow over Sphere at Mach = 0.2,
Re = 10,000 using two different time steps
Figure 9.. URANS and DES Average Surface
Pressure Distributions Compared with
Experimental Data at Various Reynolds
Numbers for flow over Sphere
Mach = 0.2, Re = 10,000s
The time history of the drag coefficient is shown
in Figures 6 and 7 and reveals important
differences between URANS and DES. The
mean value of the drag coefficient in both cases
is close to the experimentally reported value of
frequency content in each case
different. The URANS simulation appears to
damp out most of the oscillations present in the
DES run, while the DES runs show a very
chaotic oscillatory pattern quite similar to the
solutions obtained by Constantinescu et al .
Spectral analysis of the time-dependent drag
coefficient history reveals a peak corresponding
to a Strouhal number of 0.1 which is not in
agreement with the values 0.18 to 0.2 reported
experimentally .This may be due to an
insufficiently long time history sample, since
less than three full periods of this frequency are
present in our sample.
The Mach number contours depicted in Figure 8
corroborate the difference in the predicted flow
using regular Spalart-Allmaras RANS turbulence
model and detached eddy simulation (DES).
DES shows a wider range of scales present in the
flow while regular RANS models tend to smooth
out the smaller scales. This is the effect expected
from DES since the length scale redefinition
increases the magnitude of the destruction term
in the Spalart-Allmaras model, drawing down
the eddy viscosity and allowing instabilities to
develop. Predictions of the mean pressure
distribution over the surface of the sphere are
shown in Figure 9. The surface pressure is seen
to match well with experimental results at
Re=165,000 reported by Achenbach , which
is in agreement with the results reported by
Constantinescu . The average computed
separation angle of 81ocompares resonably to
experimental vale of 82.5o.
5.3 Flow over a NACA0012 airfoil.
RANS and DES were used to compute the flow
over a NACA0012 airfoil at a Reynolds number
of 105and a Mach number of 025. Runs were
performed for 10 different angles of attack
ranging from 0 to 16. Our main interest was to
study and compare RANS and DES in the stall
regime. Other studies (Shur et al. ) have
focused their interest in higher angles of attack,
such as 45, 60 or even 90 degrees, showing very
good and promising results in which DES is
within 10% of the experimental values, while
traditional RANS predict results with 40%
errors. All runs are computed as fully turbulent
to avoid having to trigger the transition point. A
mesh of 966,00 node points is used with a first
grid spacing normal to the solid boundary of 10-5
chords. The spanwise boundary conditions are
slip (inviscid)walls, and the span of the
geometry is two chords lengths.
In this case, due to time and computing
constraints, only one time step of 0.25 × c/U∞
Figure 8. Comparison of URANS and DES
Mach Contours for flow over sphere.
was used, where c is the airfoil chord. The
calculations were carried out for a maximum of
50 time units for the post-stall cases, where a
time unit represents the time it takes for the
undisturbed far-field flow to travel one chord
The time history of the lift and drag coefficient
show good agreement between URANS and
DES for angles of attack bellow 11.5 degrees.
This was expected since for pre-stall conditions
the DES model operates primarily in the RANS
mode. However, for the post-stall condition, i.e.
angles of attack over 11.5, the time history of the
force coefficients obtained using URANS and
DES showed differences similar to the ones
observed for the sphere case. The time history of
the DES results shows higher unsteadiness that
URANS, indicating that more scales are being
captured in the separated region. The DES
simulation predicts a more severe stall than the
URANS results, i.e. lower post-stall lift and
higher drag. However, both computations stall at
experimental data  is not sufficiently close to
favor agreement with one method over the other.
This is in contrast to previous work  where
DES was reported to provide closer agreement
with experiement for very high angles of attack,
such as 45 to 90 degrees.
attention was devoted to the prediction of the
onset of stall in .
Figure 10: Sample URANS Drag Coefficient
Time History for Flow over NACA0012 Wing at
Various Angles of Attack
Figure 11. Sample DES Drag Coefficient Time
History for Flow Over NACA0012 Wing at
Various Angles of Attack
Coefficient versus Angle of Attack for URANS
and DES versus Experimental Data  at two
Different Reynolds Numbers
Comparison of Computed Lift
Coefficient versus Angle of Attack for URANS
and DES versus Experimental Data  at two
Different Reynolds Numbers
Comparison of Computed Drag
A steady-state unstructured multigrid solver was
extended to unsteady flows and validated by
computing the flow over a circular cylinder.
Results show good agreement with experimental
results in Strouhal number, pressure distribution ,
and flow patterns over the cylinder.
A Detached Eddy Simulation (DES) technique
was implemented in the code and compared with
regular URANS by computing the flow over a
sphere and the flow over a NACA0012 airfoil.
The results for the sphere show that DES
provides a better description of the flow in the
separated regions. This was concluded from the
time history of the force coeffcients and from the
pressure distribution over the surface of the
sphere. The time history of the drag coefficient
computed using DES showed small scales
present in the flow that URANS did not capture.
The Strouhal number obtained does not match
the experimental value but longer time histories
are required for a more definite determination of
Prediction of the onset of stall for a NACA0012
wing was not substantially improved using the
DES approach over the URANS approach.
However, DES has been reported to provide
better agreement with experiement at very high
angles of attack, and this will be investigated in
future work. Additional work is also underway
to simulate the decay of homogeneous isotropic
turbulence in a periodic domain, in order to
calibrate the DES model for unstructured grids.
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