The 25th Computational Fluid Dynamics Symposium
Paper No. C01-3
254

Efficient implementation of WENO Scheme on structured meshesoXinrong Su, Dept. Aero. Enging, Tohoku Univ, Sendai, E-mail: su@ad.mech.tohoku.ac.jp
Daisuke Sasaki, Dept. Aero. Enging, Tohoku Univ, Sendai, E-mail: sasaki@ad.mech.tohoku.ac.jp
Kazuhiro Nakahashi, Dept. Aero. Enging, Tohoku Univ, Sendai, E-mail: naka@ad.mech.tohoku.ac.jp

In this paper the application of high order Weighted Essentially Nonoscillatory (WENO) reconstruction to the subsonicand transonic engineering problems is studied. It is observed the WENO scheme based on the characteristic variable
has better convergence speed and accuracy than the WENO scheme based on primitive variable. For engineering
problems with shock of moderate amplitude, a simplified version of the characteristic variable based WENO withwhich the CPU cost overhead can be significantly reduced is developed. Also in this work it is found for viscous case it
is better to include the viscous effect which is reflected by the modification to a parameter in the WENO reconstruction.
Numerical results indicate with the use of the simplified characteristic variable based reconstruction and the modified
parameter, the nonlinear WENO interpolation is sharply activated in the region of shock jump and meanwhile in thesmooth area the WENO interpolation weights are tuned towards the designed optimal value for better accuracy. Also
this approach demonstrates better convergence rate than the primitive variable based reconstruction. Several practicalcases are calculated to demonstrate the accuracy and efficiency of the current methodology.

1. Introduction
It is well known computational fluid dynamics (CFD) offers theability to perform predictions of the full field vehicle flows at low
cost compared to wind tunnel experiments. While the accurate
and fast-turnaround simulation still demands new algorithms. Up
to now, the widely used codes for engineering problems employ
second-order schemes. Higher order numerical methods, such as
the Weighted Essentially Nonoscillatory and Discontinuous
Galerkin (DG), can provide more accurate solutions with fewer
mesh points. In the higher order schemes, the numericaldissipation is reduced and this is one of the key elements ofaccurate capture of detailed flow structures, such as tip vortex andshock-vortex interaction. Currently most of the efforts are exerted
in the application of these methods in the field of high-fidelity
simulations, such as large eddy simulation (LES) andaeroacoustics. While high order schemes are rarely applied toengineering problems, such as wing-body and turbomachinery
cases. Compared to the lower order numerical schemes, highaccuracy schemes are CPU extensive. Due to the more stencils
needed for high accuracy and reduced numerical dissipation,these methods are always numerically stiff and usually exhibit
slower convergence speed and reduced stability.
Of all the high order schemes, Essentially Nonoscillatory (ENO)
and WENO algorithms can be used to replace the widely used(1)
Monotone Upwind Scheme for Conservation Laws (MUSCL)
approach in structured codes for more accurate inviscid fluxes. In
the ENO approach, several candidate reconstruction stencils areformed and the nonoscillatory property is achieved by selecting
the one with maximum smoothness. In this approach the stencil
interpolated across the discontinuity are never used and theoscillations are prevented. Based on the ENO algorithm, Liu et(2)
al. developed the WENO scheme by the convex combination of
all the stencils to improve the accuracy. In the WENO algorithm,
the stencil interpolated across the discontinuity is assigned a nearzero weight and thus the nonoscillation property is preserved.
(3)
Jiang and Shu further developed the WENO algorithm bydevising a new set of weights for higher accuracy. Currently the
WENO algorithm is widely used in the high fidelity simulations.

Especially in the simulation of shock-turbulence interactions, it
serves as the building part to capture the shock without dissipating(4)(5)
the small scale turbulent fluctuations . Ren et al. developed a
Roe type characteristic-wise hybrid compact-WENO scheme inwhich compact difference is used in the smooth regions and theWENO scheme is activated near the shock by a weight function.
(6)
Zhang and Shu proposed a new smoothness indicator to
improve the convergence issue of WENO scheme to steady state.
Compared to the extensive researches and applications in highfidelity simulations, the use of WENO scheme in the engineering(7)
problems are less studied. Nichols et al. applied the fifth order
WENO in the NASA OVERFLOW code and tested the schemewith problems involving vortical flows, strong shock and large
scale unsteady flows. The WENO scheme was found to provide
much lower numerical dissipation and dispersion compared to the(8)widely used third order MUSCL reconstruction. Shen et al.
studied the weights stability and accuracy of fifth order WENO
scheme. It is found the weight of the original WENO scheme maylead to oscillation even for smooth flows. According to numericalexperiments, it is proposed to increase a parameter fromtoand the convergence property and the accuracy can besignificantly improved.
In this paper the authors are interested in how to efficiently
apply the WENO reconstruction to engineering problems for better
accuracy. Also the numerical method should have favorable(7,8,9)convergence speed for steady state solutions. In the literature,for engineering problems using either WENO or MUSCL typereconstruction, it is believed there is no need to resort to theexpensive characteristic variable based reconstruction. While inthis paper it is found the use of characteristic variable helps toimprove the accuracy and convergence speed. Also for thetransonic problems interested in this paper, a simplified version
can be used to obviously reduce the cost overhead in thecharacteristic variable based approach. The application of WENO
scheme can be further improved by taking into account of theviscous effect. With the use of characteristic variable and theconsideration of viscous effect, the current implementation tends
to sharply activate the nonlinear interpolation near the shock for
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The 25th Computational Fluid Dynamics Symposium
Paper No. 254
interpolation is a nonlinear process. In the smooth area, all theweights approach the designed optimal values and linear
interpolation is used indeed.
There are several choices of the variable to be reconstructed,
such as the primitive variable, conservative variable, or thecharacteristic variable. Currently in most of the engineering cases
using the WENO algorithm, the WENO reconstruction is used(8)
either to the conservative variables , or to the primitive(7,9)
variables . As conservative variable and primitive variable arealways available in the code and thus these methods are
straightforward. Another approach is to use the WENO method in
the characteristic field. Let
be an unit length vectorof the surface normal, the Jacobian matrix is defined as

monotone solution. At the same time in the smooth area theunnecessary nonlinear interpolation is suppressed. Also thecurrent implementation has similar or better convergence rate
than the primitive variable based WENO. Several practicalengineering problems are used to validate the accuracy andconvergence issues of the current method.
2. Numerical methods
The governing equation can be written in the Cartesian
coordinate asin whichdenotes the conservative
variable.
denote the convective fluxes and also include
viscous fluxes for viscous cases. Also the primitive variable
is widely used in this work. The governing
equation is nondimensionalized with the freestream conditions
and proper length scale.

, also denotesandthe right and left eigenmatrix of theabove matrix, respectively. In the calculation ofand, the conservative variables are multiplied by the left
eigenmatrix

to get the characteristic variables

WENO reconstruction is used to calculateand.

The last step is the transformation from characteristic variable toconservative variable byFig. 1 Reconstruction stencils for the fifth order WENO algorithm
The reconstruction of variables with WENO method provides amethod to achieve higher order accuracy both in the smooth partand in the shock part of flow region, compared to the currently
widely used MUSCL approach. In the WENO method, theessentially non-oscillation and high order accuracy properties are
realized by the convex and nonlinear combination of several loworder approximations. As demonstrated in Fig. 1, the fifth-orderaccurate WENO reconstruction of. Then the. Compared to the methods based on the conservative variableand primitive variable, in the characteristic variable based
approach, the left and right eigenmatrices have to be formed andalso matrix-vector multiplications are needed. These result in
obvious CPU cost overhead. Also in the literature it is believed forengineering problems with high order reconstructions, such as thefifth order WENO scheme, the using of characteristic variable
does not result in additional gains. As a result it is seldom used inengineering problems for these two reasons.
In the following the characteristic variable will be computed fromprimitive variable, as the transformations are much simplified
compared to from conservative variable. The governing equation
can be transformed into

can be written as

where. The Jacobian matrix and the eigenvalue decomposition are
defined as

and the weights are defined as(1). In the above equation, the smoothness indicators

and up-to-now the calculation ofare. For clarity, two dimensional case will be discussed here.
Denoting
a unit vector which is perpendicularto, the left eigenmatrix can be described in theform ofis completed andcan be constructed in the way symmetric toand will not

be detailed here. In the shock region, the reconstruction stencil
across the shock will be assigned a small weight and now the. As shown in Eqn. (1), in the WENO algorithm, the smoothness isestimated by the variations of the variables to be reconstructed.
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The 25th Computational Fluid Dynamics Symposium
Paper No. 254
components corresponding to the vorticity waves can be omitted
in the WENO reconstruction.
As a result for these cases a simplified version of thecharacteristic variable based WENO reconstruction can be
introduced. Denoting
as the j-th row ofandthe j-thcolumn. For both two and three dimensional problems, thenonlinear WENO interpolation is needed only for the first threecomponents of characteristic variable, while linear reconstruction
is used for the remaining part. The reconstruction is executed inThus looking at the characteristic variable based WENO algorithm,
has the form of(2). The first component ofrepresents the entropy wave
and the second and the third component represents thedownstream and upstream propagating pressure wave,
respectively. The last one represents the shear wave, viz., vorticity.
Compared to the primitive variable or conservative variable, thecharacteristic variable better reflects the flow features. For
example in the isentropic flow,. So if characteristic
variable based WENO reconstruction is used, according to thedefinition of the WENO weights in Eqn. (1), linear interpolation
instead of the lower order nonlinear interpolation will be used inthis characteristic field. While for the same case, due to thevariations of and , the primitive variable and the conservative
variable based approaches tend to activate the low ordernonlinear interpolations despite the shockless flow.

the following steps: (1). Compute

from

and

and then calculate. (2). Primitive variable istransformed into characteristic form by
and WENO reconstruction is used to computeand. (3). Denoting the remaining part as, linear reconstruction is used for this part.
For third order accuracy, the reconstruction has the form of. For fifth order accuracy the reconstruction can be expressed as

Fig. 2 Case with shock wavealigned with the gridFig. 3 Case with shock wave
not aligned with the grid(4). The last step is to get

andby summing up

the two parts
Obviously the first three components in Eqn. (2) varysignificantly across the shock, while the variation of the lastcomponent is a bit different. As demonstrated in the Fig. 2, in thecase of the shock wave is aligned with the grid, according to the
Rankine-Hugoniot conditions, the velocity difference in thetangential direction vanishes. Thus the last component
approaches zero and at least there is no need to use the nonlinear
WENO interpolation to suppress the spurious oscillation in thischaracteristic field. In the case the shock wave is not aligned withthe grid lines, as depicted in Fig. 3, the velocity differencetangential to the shock wave is zero and does not contribute to. Denoting the velocity component normal to the shock wave
as, in this casecan be expressed as

Compared with the original implementation of characteristic
variable base WENO reconstruction, the current simplified version
obviously yields reduced computation cost. As only
andinstead of the full matrices
have to be formed and also the count of matrix-vector
multiplications used to calculate the characteristic variable has
been reduced. And also the costly WENO algorithm is needed
only for the first three components of the characteristic variable,
instead of all the four components in two dimensions or fivecomponents in three dimensions.
For transonic flows in the engineering regime, there is a slightly
rise of vorticity magnitude across shock wave, while the vorticity
increases rapidly in the region with high velocity shear, such as in
the boundary layer and the tip vortex region. The requirement of
accurate tracking vorticity along long distance is necessary inmany engineering problems. While due to the inherent numericalviscosity in numerical schemes, the vorticity is always dissipated(9)much earlier . In the current simplified WENO algorithm, linear
reconstruction is used for the components corresponding to thevorticity waves, thus the numerical dissipation is further reduced
and better accuracy for vortical flows is possible.

For the engineering problems interested in this paper, the shock
wave is of moderate amplitude and according to the
Rankine-Hugoniot conditions
is a limited value. Also in
the generation of structured mesh, the mesh lines are always
positioned to be aligned with the dominate flow features, such as
shock waves and boundary layers. Thus for high quality meshis always a small angle. So in the case the shock wave is notaligned with the grid lines, for transonic flows calculated withstructured meshes, there is always no need to use the nonlinear
and also expensive WENO interpolation to the component of thecharacteristic variable corresponding to the vorticity wave. For
three dimensional problems similar analysis also applies and will
not be repeated here and in three dimensions the two3

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The 25th Computational Fluid Dynamics Symposium
Paper No. 254
algorithm by the modification to . First the Reynolds number
based on the local mesh spacing is computed and ifislower than a pre-defined threshold, the flow region near this part isdenoted as diffusion-controlled and linear reconstruction instead of
the WENO algorithm is used. In the current work the threshold
Reynolds number is taken to be 5. Then for the rest of the domain,is re-defined as

The WENO reconstruction algorithm computes several
low-order approximations ofandand the final

value is obtained by the weighted summation of all the low-orderapproximations. This process is nonlinear and the weights are
directly related to the smoothness indicators. As seen in Eqn. (1),the weight is determined by the ratio of its smoothness relative tothe total smoothness. Thus there may be situation in which theabsolute variation of is small, while the relative variation of isbigger. In this case the reconstruction stencil with relatively large
variation will be assigned a small weight, no matter whether theshock wave do exists. Also during the transient stage from initialsolution to the final converged solution, there is always small
amplitude variation even in the subsonic area, in this case thesmoothness indicators will work unreasonably and this may hinder
the time marching towards convergence. According to Eqn. (1), if, the weights always approach the designed optimalvalues despites of the smoothness indicators. Thus although thesmall value
in Eqn. (1) is originally designed to avoid thedenominator from zero, in real-world problems,
acts as a
threshold to distinguish the acceptable small amplitude variationfrom the shock wave. In the smooth area, it is better to have on
the same order or bigger thanso the designed order ofaccuracy can be achieved, while near the shock should bemuch smaller thanto activate the shock capturing. In theoriginal WENO algorithm, is set to. In the work of Shen et(8)al. , according to numerical experiments, this value is increasedto. It is hard to define a value suitable for broad range offlows, while it is believed using a bigger which also does not
destroy the shock capturing ability is beneficial for both accuracy(8)
and convergence. Similar as in the work of Shen et al. , in thispaper is set to.
Note that for smooth flows
should converge to zero withmesh refinement; while for the reconstruction stencil across theshock wave,
converges to a finite value with meshrefinement. So it is better to have the parameter to be functionof the local mesh spacing. For general cases it is a hard job and in
this work the effect of local mesh spacing is considered in a more
physical manner. In all the above discussions, only the inviscid
part in concerned. For viscous flows, the viscosity, including themolecular part and the turbulent part, helps to stabilize thecalculation and smooth the flow field. So taking into account of thephysical viscosity would better suit the physical process. For
simplicity, looking at a one dimensional convective-diffusion
equation. With the use of, in domains dominated by viscous
effects, the admissible variations is enlarged and the WENO
weights tend to the designed optimal value and higher accuracy
can be achieved. For high Reynolds number flows, at least in theviscous sub-layer, no WENO reconstruction is used andreconstruction order is forced to be the designed value and thiswould yield better accuracy for the boundary layer and this also
results in a bit cost saving by the way. In the off-body part of theflow, although the turbulent viscosity is much bigger than themolecular viscosity, the mesh spacing is much larger, thus in thispart
is seldom a small value and the physical viscosity
has limited ability to suppress the oscillations. While on the other
hand with the development of computation resources, larger scaleand denser mesh is used to resolve the flow problem, so theis decreasing with denser mesh and physical viscosity
has increasing capability of suppressing oscillations. It has to be
noted there maybe a better formulation to bring in the viscous
effect and the approach in this work is just heuristic.
With the calculatedand, throughout this(10)

paper the HLLC Riemann flux is used to compute the inviscid
flux across the cell interface. For viscous cases, second orderscheme is used to discrete the viscous term. For turbulent cases,(11)
the two equation Menter-SST turbulence model is used to
calculate the turbulent viscosity. Unless otherwise stated, thecomputation throughout this paper is conducted with multiblock
body-fitted structured mesh and Alternating Direction Implicit (ADI)
method with three levels of multigrid are used for the time
integration.
3. Numerical example: NACA 0012 airfoil
The NACA-0012 airfoil case is solved with the Building Cube
(12,13)
Method. In the Building Cube Method block structured
Cartesian mesh is used to discretize the flow region and all theblocks have the same number of points. Ghost cells are used to
exchange information between neighbor blocks. This test case isconducted in the two-dimensional inviscid mode and within thecurrent Building Cube Method, ghost cell based Immersed(14)
Boundary method is used to treat solid walls not aligned withthe mesh lines. Fifth order WENO method is used for the inviscid
fluxes. A five stages explicit Runge-Kutta time marching method
and the Full Approximation Scheme (FAS) type multigrid method
are used to integrate the governing equation towards
convergence.
First the case with inflow Mach number
is tested.
For this case the shock wave is approximately aligned with themesh lines. Both the primitive variable based WENO and thecharacteristic variable based WENO are computed. Also in orderto test the sensitivity of the convergence to the small number ,. This equation is semi-discretized as. Thus a Reynolds number based on the local mesh spacing,
can be defined. It can be easily proved if, thenwill be monotonically increasing or
decreasing. In this case the viscosity is enough to dissipate theunphysical oscillations. Inspired by the analysis, the physicalviscosity is introduced into the current implementation of WENO
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The 25th Computational Fluid Dynamics Symposium
Paper No. 254
shock wave is sharply and monotonically captured, while there areentropy oscillations with the primitive reconstruction.
Another case with the incoming Mach number
issimulated with both the characteristic and primitive variable based
reconstruction. The small number has two choices,and. Also the simplified WENO reconstruction withistested. In this case the shock wave is not aligned with the meshlines and the shock extends further into the ambient and a muchlarger computational domain is used. The comparison of
convergence histories is given in Fig. 7. As shown in Fig. 7, only
the two characteristic variable based reconstructions with
reduce the residuals smoothly and the other three
result in convergence stall. While even withthecharacteristic WENO has better convergence than the primitive
counter part.

both
and
are tested. The comparison of
convergence histories are given in Fig. 4. Of all the four WENO
reconstruction strategies, for both choices of the small value , thecharacteristic variable based WENO method yields smoothconvergence. For the primitive variable approach, withthe convergence speed is similar as that for the characteristic
based approach, while withthe residual has been
reduced about three orders of magnitude and then fails toconverge any more.

Fig. 4 Comparison of convergence histories for various WENO
strategies for the NACA-0012 case,In order to compare the characteristic and primitive strategies,
the WENO weights used to calculate

in the region

contains the shock wave are compared. According to thecomparison, in the characteristic variable based reconstruction,
only the component corresponding to the downstream
propagating pressure wave demonstrates large variations. The
first and the third component yield moderate variations across theshock. In this case the shock wave is approximately aligned withthe mesh lines, thus the component related to the transverse
velocity has minor change across the shock and the WENO
weights for the fourth component approach the designed value.
The simplified version of characteristic variable based WENO isthus reasonable and the component representing the vortex wave
can be linear interpolated without harming the shock capturing. In
the primitive variable based reconstruction, the nonlinear process
selecting the smoothest candidate stencils are activated at moregrid points and also all weights demonstrate larger variations, inother words, the primitive variable approach tends to apply lower
order reconstruction to more grid points. The entropy distributions
are compared in Fig. 5 and Fig. 6.

Fig. 5 Entropy distribution, thecharacteristic variable based
reconstruction

Fig. 7 Comparison of convergence histories for various WENO
strategies for the NACA-0012 case,
The computational results are compared in Fig. 8 and Fig. 9.
The simplified characteristic variable based WENO sharply
captures the shock wave. In this case the angle between theshock front and the mesh line is about, while from theconvergence history and the density contour, the simplified
version still works well and yields similar results as the original
version. A significant difference can be observed that thenumerical result obtained with primitive variable based WENO
contains post-shock oscillations while the oscillations in the former
two methods are much smaller. Based on the numerical resultsfrom the NACA-0012 case, the characteristic variable based
WENO reconstruction has better convergence speed andaccuracy, thus should be preferred.

Fig. 6 Entropy distribution, theprimitive variable based
reconstruction

Fig. 8 Density distribution, thesimplified characteristic variable
based reconstruction

With the characteristic reconstruction the entropy rise across theFig. 9 Density distribution, theprimitive variable based
reconstruction

4. Numerical example: RAE 2822 airfoil
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The 25th Computational Fluid Dynamics Symposium
Paper No. 254
they have different convergence properties. The computation
results with the primitive variables based WENO are shown in Fig.
13 and the black dot in every picture denotes the grid point withmaximum residual. In two dimensions there are two values ofand the minimum of the two are shown in Fig. 13. The
turbulent viscosity reaches its maximum near this point and thus
has a rather small value of 50. Another set of numericalexperiments are conducted to examine the effect of bringing inviscous parameter into the WENO reconstruction. Bothandare tested. Also the calculations withand without the viscous correction are conducted.

To examine the performance of the WENO reconstruction fortransonic and turbulent flows, the RAE2822 airfoil is simulatedwith fifth order WENO reconstruction. In this case, the meshconsists ofgrid points. The freestream condition
considering wind tunnel correction isand the Reynolds number is.
With, three sets of numerical experiments have been
conducted, including the characteristic variable based, theprimitive variable based and the simplified characteristic variable
based WENO reconstructions. Note that the
basedcorrection is absent in these three computations. As shown in Fig.
10, similar as in the NACA-0012 case, forall the three
WENO reconstructions demonstrate smooth convergence. To
further examine the behavior of these reconstruction methods
is set to a stringent value ofand the residuals are shown in
Fig. 11. In this situation, both characteristic variable based WENO
methods successfully converge with minor loss of speed. While
the residual of primitive variable based WENO fluctuates at thelevel of about.

Fig. 14 Convergence histories
with and without viscous effectin the primitive variable based
WENO,

Fig. 10 Comparison of
convergence histories ofvarious WENO strategies,

According to the convergence histories in Fig. 14, with the effectof, the convergence behavior has been greatly improved
as in this situation the effective value of is significantly magnified.
For, taking into account of
has little effect on
the convergence, while the accuracy has been obviously
improved. The WENO reconstruction weights ofcorresponding to the velocity-Y component are compared in Fig.
15. For both choices of , the weights are tuned towards thedesigned value for optimal order of accuracy. Note that there arestill some weights not equal to the designed value and the viscous
effects can be further strengthened to deactivate the usage of thenonlinear interpolation for highest accuracy in this region, while atthe risk of worse shock capturing and a better option is to increase
the mesh resolution in the wake region.

Fig. 11 Comparison of
convergence histories ofvarious WENO strategies,

By carefully monitoring the residuals at all mesh points, it isfound the convergence degradation takes place in the wakeregion. In this area the Mach number is about 0.5 and there arelarge variations of both primitive and characteristic variables andthe mesh spacing is much larger than in the boundary layer. As
analyzed in the above, in this case the physical viscosity is notenough to fully damp the variations. While in this case thecharacteristic variable based approach behaves better and still
converges successfully. From the NACA-0012 case and the
RAE2822 case the convergence speed of the characteristic
based method is better and less sensitive to the small value .

Fig. 16 Entropy distributions,simplified characteristic variable
based WENO

Fig. 12 Comparison of
surface pressure coefficients,

Fig. 15 WENO reconstruction
weights along the I=const meshlines in the wake region,forthe velocity-Y component

Fig. 17 Entropy distributions,primitive variable based WENO

For characteristic variable base WENO, the situation is similar
as in the primitive variable based WENO with
. The
convergence speed is less affected, while all the reconstruction
weights are tuned towards the optimal value and thus theaccuracy is improved in the wake region. With the viscous effect,
the order of accuracy of WENO reconstruction in the boundary
layer and the wake region is primarily determined by the viscousparameter. In the shock region, as the viscosity is in
general much smaller, thus the WENO reconstruction weights still
depend on the small value . Similar as in the NACA-0012 case,Fig. 13 Computed Reynolds
number based on the local
mesh spacing

The surface pressure coefficients are compared in Fig. 12. All
the results are in good agreements with the experiment, though
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The 25th Computational Fluid Dynamics Symposium
Paper No. 254
stalls. With the simplified characteristic variable based
reconstruction, the residual can be further reduced to about.

the characteristic variable based WENO has better accuracy andthe primitive variable based WENO tends to activate the nonlinear
interpolation at more grid points. The entropy distributions arecompared in Fig. 16 and Fig. 17 and the characteristic variableapproach has obviously better accuracy.
5. Numerical example: Rotor-37 compressor
The third test case is an example of the three dimensional flow
through the transonic compressor NASA Rotor-37. This
compressor rotor is widely used to test the numerical algorithms
and to study the complex flow phenomena of compressor
aerodynamics. Plentiful of flow features important to theengineering community co-exist in the Rotor-37 case, such as
strong shock wave-boundary layer interaction induced flow
separation, tip vortex and vortex-shock wave interactions. Thus
the capturing of detailed flow features demands accurate
algorithms. On the other hand because of the strong adverse
pressure gradient and the complex flow structures in this case, it ishard to guarantee the convergence and stability of high orderschemes. So in the literature central scheme with artificial
dissipation, also third order MUSCL reconstruction are mostly
used in this case and fifth order WENO type high order scheme isseldom tested.
In this work the fifth order WENO reconstruction is tested, bothin the primitive based and the simplified characteristic based styles.
The small number is set toand the viscous effect isincluded. The grid used in this case is a multiblock structured
mesh and consists of approximately 1M points. The spacing
between the solid and the first off-wall point is tuned to guarantee
theis roughly 1. One-dimensional distribution of total
temperature, total pressure and the flow angles at the inlet surface
are treated with the use of 1D Riemann invariants. At the outlet
surface, static pressure distribution based on the radial equilibrium
equation is used.
Primitive
Characteristic
Characteristic
Ordervariablevariablevariable, simplified
3
1.00
1.78
1.37
5
1.62
3.28
2.06
Table. 1 CPU cost of various WENO strategies

Fig. 18 Convergence histories of fifth order WENO
reconstruction for the Rotor-37 test caseIn order to assess the actual reconstruction order of the WENO
algorithm in practical engineering problems, a parameter in theform ofis used to reflect the deviation of the interpolation weight from its
designed optimal value. The larger of this value, the lower
reconstruction order of accuracy is used. The distributions ofat the radial position of
blade height are given in Fig. 19 andFig. 20.

Fig. 19 Distribution of,
simplified characteristic variable
based WENO

For this three dimensional case, third order WENO and fifth
order characteristic variable based WENO are also executed forseveral steps to assess the CPU cost of various WENO strategies,
as shown in Table. 1. Note that the CPU time for third orderprimitive variable based WENO reconstruction is set to be thereference value and also this CPU time is not the total CPU time,
but the one elapsed in the reconstruction step. From thecomparison, the simplified version of characteristic based WENO
developed in this work dramatically reduces the CPU cost. For ageneral purpose solver, there are moreover several
CPU-extensive procedures, such as the time marching, Riemannflux and the solution of turbulence equation. Thus accompaniedwith the CPU time reduction comes from the simplified version,
the total CPU cost overhead is aboutcompared to theprimitive variable based approach.
The convergence histories of both WENO strategies arecompared in Fig. 18. For the primitive variable based approach,the residual is reduced fromto about
and thenFig. 20 Distribution of,primitive variable based WENO

From Fig. 19 the nonlinear interpolation process is activatedonly in the regions of leading edge shock and the passage shockin the simplified characteristic variable based WENO. While asshown in Fig. 20, besides the shock region, the nonlinear
interpolation is used at more grid points in the primitive variableapproach. An interesting phenomenon is in the region from trailingedge to the outlet in Fig. 20, the use of nonlinear interpolation issuppressed and it is attributed to the inclusion of viscous effect, asin this region the turbulent viscosity is much larger andisbetween 10 and 100.
The vorticity magnitudes based on the relative velocity at theaxial positions ofchord andchord are compared inFig. 21 and Fig. 22. At thechord, both methods yieldapproximately the same tip vorticity, while at downstream position,
the vorticity is dissipated quicker in the primitive variable based
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The 25th Computational Fluid Dynamics Symposium
Paper No. 254
approach, especially the suction-side tip vortex.

Fig. 23 Convergence historiesof the DLR F4 wing-body caseFig. 24 CD convergence of
the DLR F4 wing-body case

The comparison of the isosurface withis given in Fig.
25 and Fig. 26. From the isosurface in Fig. 25, in the simplified
characteristic variable based reconstruction, the nonlinear
interpolation is activated near the windshield, the leading edge of
the wing and trailing edge and also in a small part of the tip vortex.
While a significant difference can be observed in the far-field. In
the primitive variable approach the nonlinear interpolation isextensively used in the tip vortex region and the wake region.
Thus although both methods yield similar lift and drag coefficients,
the simplified characteristic variable based reconstruction better
preserve the tip vortex along long distance because in this region
highest order linear interpolation is always used. Thus for cases(9)
the tip vortex is important, such as the helicopter flows , using
characteristic variable would help to further improve the accuracy.

Fig. 21 Vorticity magnitude distributions at axial positions ofchord andchord, simplified characteristic variable based
WENO

Fig. 25

isosurface, simplified characteristic variable
based WENO

Fig. 26

isosurface, primitive variable based WENO

Fig. 22 Vorticity magnitude distributions at axial positions ofchord andchord, primitive variable based WENO
The evolution and the breakdown of the tip vortex are importantto the stability of the compressor and thus the accurate prediction
is necessary. From the comparison the characteristic variable
based WENO reconstruction gives better accuracy and efficiency
and should be preferred.
6. Numerical example: DLR F4 wing-body
The following test case is the DLR F4 wing-body configuration.
The flow conditions used isand theReynolds number
based on the wing's mean
aerodynamic chord length. The mesh consists of about 3.3M grid
points and is provided by the workshop committee. Note that thisstandard mesh is of questionable quality with high skewness inthe near-wall region.
Fifth order WENO reconstruction with both the simplified
characteristic variable and primitive variable based reconstructions
are used. Also thebased viscous correction is used toreflect the viscous effect in the reconstruction.
For this case, as given in Fig. 23, the WENO reconstruction
gives reasonable convergence speed, though the grid has a low
quality. The two reconstructions have similar convergenceproperties and give the same lift coefficients. While the dragcoefficient using the primitive variable based approach is about 1.5
counts higher.

7. Conclusions
In this work the application of high order WENO scheme toengineering problems is studied. In the literature it is believed forengineering problems, the use of characteristic variable in the highorder WENO reconstruction has little superiority, while at thegainless cost of CPU overhead. Thus for engineering problems
the primitive variable or conservative variable is always used. In
this paper, the choice of variable is found to do play a role in termsof accuracy and convergence. Also the CPU time overhead canbe obviously reduced based on some reasonable simplifications.
From the work in this paper the following conclusions can be
made:(1). For engineering problems, the characteristic variable based
WENO has better accuracy and the nonlinear interpolation
procedure is always sharply activated near the discontinuities.
While in the primitive variable based approach it tends to activate
the nonlinear interpolation at more grid points and thus thereconstruction accuracy is lower. As a result the characteristic
8

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The 25th Computational Fluid Dynamics Symposium
Paper No. 254(1994), pp. 598--1605.
(12) Kazuhiro Nakahashi, “High-density mesh flow computations
with pre-/post-data compressions”, AIAA-2005-4876, 2005.
(13) Kazuhiro Nakahashi, Aya Kitoh, Yuta Sakurai, and Matthias
Meinke, “Three-dimensional flow computations around an
airfoil by building-cube method”, AIAA-2006-1104, 2006.
(14) Rajat Mittal and Gianluca Iaccarino, “Immersed boundary
methods”, Annual Review of Fluid Mechanics, 37 (2005), pp.
239--261.

variable based WENO has better capability of monotone captureof shock wave and better preserves detailed flow features.
(2). For engineering problems in the subsonic and transonicregimes, the CPU overhead of the characteristic based WENO
can be obviously reduced. Based on the Rankine-Hugoniot
conditions, the components of characteristic variablecorresponding to vortex wave can be linear interpolated without
visible loss of shock capturing capability.
(3). The characteristic variable based WENO has better
convergence issue and is less sensitive to the small number .
The improved convergence speed can somewhat counteract the
CPU time overhead in the characteristic variable based WENO.
(4). For viscous case, it is preferred to take into account theviscous effect. With modification to the small number , in theviscous dominated region, the reconstruction order is tuned
towards the designed value. Also the convergence speed can be
improved with the consideration of viscous effect.
8. Acknowledgement
This work was supported by JSPS KAKENHI(21226018).
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