# Presentation¶

Since 1997, the elsA software package is the Onera multi-purpose tool for applied aerodynamics and multiphysics, which capitalizes on the innovative results of Computational Fluid Dynamics (CFD) research over time. The elsA CFD simulation platform solves the compressible Navier-Stokes equations and deals with internal and external aerodynamics from the low subsonic to the high supersonic flow regime. A very wide range of aerospace applications is covered by elsA: aircraft, helicopters, tilt-rotors, turbomachinery, counter-rotating open rotors, missiles, unmanned aerial vehicles, launchers... elsA is also used for configurations outside aerospace field, such as car turbochargers, steam turbines and wind turbines. The development and the validation of the elsA software benefit by input from research partners and feedback from industry users. In particular, elsA is used as a reliable CFD tool by Airbus for transport aircraft configurations, by Safran group for turbomachinery flow simulations and by Airbus Helicopters for helicopter applications. Verification and validation of a multi-purpose software including numerous CFD capabilities are very timeconsuming. Community benchmarks allow a reduction of this burden since they mutualize part of the effort by providing reference experimental data and meshes. They also offer very informative code-to-code comparisons. elsA results are presented on configurations from the Turbulence Modeling Resource (TMR) website supported by NASA Langley (http://turbmodels.larc.nasa.gov) and compare with experimental data and CFL3D code.

# Academic-2DAxiNear-SonicJet¶

- 2D-Axi configuration
- Steady subsonic flow
- Turbulent viscous flow
- Perfect gas fluid model
- Van Leer upwind fluxes
- Van Albada limiter
- Gradients calculated on cell centers then corrected on interfaces
- Implicit scalar LU-SSOR
- Time integration : Backward Euler
- Roe upwind flux for transport equations
- Spalart-Allmaras turbulence model
- CGNS entries
- Iterations number=10000
- CFL=3.000000
- Harten isentropic correction coefficient (turbulent system)=0.010000

# Academic-2DAxiSubsonicJet¶

- 2D-Axi configuration
- Steady subsonic flow
- Turbulent viscous flow
- Perfect gas fluid model
- Van Leer upwind fluxes
- Van Albada limiter
- Gradients calculated on cell centers
- Implicit scalar LU-SSOR
- Time integration : Backward Euler
- Roe upwind flux for transport equations
- Spalart-Allmaras turbulence model
- CGNS entries
- Iterations number=10000
- CFL=3.000000
- Harten isentropic correction coefficient (turbulent system)=0.010000

# Academic-2DConvexCurvatureBoundaryLayer¶

## Case presentation¶

The purpose here is to provide a validation case for turbulence models. Unlike verification, which seeks to establish that a model has been implemented correctly, validation compares CFD results against data in an effort to establish a model’s ability to reproduce physics. A large sequence of nested grids of the same family are provided here if desired. Data are also provided for comparison. For this particular “essentially incompressible” curvature case (from Smits, Young, and Bradshaw), the data are from experiments.

The experiment utilizes a constant area square duct of height 0.127 m with a rapid 30 degree bend (inner radius of curvature is 0.127 m). In the experiment, the aspect ratio of the duct was 6:1.

The primary focus of this case is to assess turbulence models for convex wall curvature (the lower wall in this case). The concave wall destabilizes boundary layers, increasing turbulence levels and the thickness of the boundary layer. In concave curvature, Gortler vortices generally form. These vortices tend to be quasi-stable, and can result in steady or slowly varying, large scale spanwise variations in boundary layer. The measurements from the Smits case on the concave wall show significant spanwise variations in the skin friction, consistent with the presence of Gortler vortices. We provide curves for the minimum and maximum skin friction along the concave wall from the test. Given the nature of streamwise flow in concave curvature, interpretations of differences between CFD and test in this region should take into account both the uncertainty in the test results and the difficulty of capturing Gortler vortices in a steady state computation in 3-D, and the impossibility of capturing their effects in 2-D. For this test case the duct is modeled in 2-D. Results on the concave wall are provided for guidance and comparison between models, but should not be interpreted as a definitive discriminator between models.

The reference velocity (Uref) near the inlet is 31.9 m/s. The back pressure is chosen to achieve the desired flow. The upstream “run” length is chosen to allow the fully turbulent boundary layer to develop naturally, and achieve approximately the correct boundary layer thickness upstream of the bend. The upper and lower boundaries are modeled as adiabatic solid walls. The following plot shows the layout of this case, along with the boundary conditions. “Pt” refers to total pressure, “P” refers to static pressure, and “Tt” refers to total temperature.

Some of the experimental data for this case will be shown below. The profiles of interest are at s= -0.185 m (x=-0.166124 m) upstream of the bend, and x= 0.030, 0.183, 0.335, 0.635, and 1.250 downstream of the bend. Data are given in local coordinates aligned with the inner duct wall and wall normal.

The experimental data reference is: Smits, A. J., Young, S. T. B., Bradshaw, P. “The Effect of Short Regions of High Surface Curvature on Turbulent Boundary Layers,” J. Fluid Mech., Vol. 94, Part 2, 1979, pp. 209-242. The experimental data are available on this site from:

Note that some inconsistencies were discovered in the experimental data as posted from Oct 30, 1981, in comparison with JFM (1979). The data were revisited with the help of the original author (Lex Smits) during Oct-Nov 2012. The corrected data are provided. Note that skin friction coefficient values are given here based on Uref, for convenience when comparing against CFD results. However, the original experimental data files give Cf based on local edge velocity Ue.

## Results of current version¶

A series of 5 nested 2-D grids are provided for the Convex Curvature case. Each coarser grid is exactly every-other-point of the next finer grid, ranging from the finest 1025 x 385 to the coarsest 65 x 25 grid. The finest grid has minimum spacing at the wall of about y=1.1 x 10-6, giving an approximate average y+ of less than 0.1 at the Reynolds number run. The grid is stretched from both walls in the wall-normal direction, and there is some stream-wise clustering near the bend. The figure below shows a near-field view of the 513 x 193 grid. Results were performed on 513x193 grid.

Computations were performed on the 513x193 grid; results are those of elsA v3.6.04.

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

AUSMP+ Miles upwind scheme

Van Albada limiter

Gradients calculated on cell centers then corrected on interfaces

Implicit scalar LU-SSOR

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 1 coarse grid(s)
- 1 iterations on coarse grid
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model , Rotation/Curvature Correction

CGNS entries

Iterations number=20000

CFL=100.000000

Harten isentropic correction coefficient (turbulent system)=0.010000

# Academic-3DBumpInChannel¶

## Case presentation¶

The purpose here is to provide a large sequence of nested grids of the same family, along with results from existing CFD codes that employ specific forms of particular turbulence models, in order to help programmers verify their implementations of these same models. On a given grid, there may be differences between the results from different codes, but presumably as the grid is refined the results should approach the same answer (if the flow conditions and boundary conditions are the same). With verification, the purpose is not to establish the “goodness” of a model compared to experiment, but rather to establish that a model has been implemented correctly, as intended according to the equations and boundary conditions. (It is through validation that a model’s “goodness” is established.) The purpose here is primarily verification.

The 3D bump-in-channel case is a three-dimensional version of the 2D bump-in-channel verification case, with spanwise variation added. In this 3D case the z direction is “up” and y is “spanwise.” It was run at M = 0.2, at a Reynolds number of Re = 3 million based on length “1” of the grid. The body reference area is 1.5 units. This lower wall is a curved viscous-wall bump extending from x=0 to 1.5 at the two sides of the computational domain y=0 and y=-1, but starting and ending further downstream at y locations inbetween. The maximum bump height is 0.05. The “2D” definition of the bump at the y=0 plane is:

for along

and

But the x-location of any position on the bump varies in the spanwise direction between y=0 and y=-1 according to:

for

where x0 is any given x-location of the “2D” shape at y=0, and The upstream and downstream farfield extends 25 units from the viscous-wall, with symmetry plane BCs imposed on the lower wall between the farfield and the solid wall. The upper boundary is a distance of z=5.0 high. It is taken to be a symmetry plane. The left and right walls are also taken to be symmetry planes. The following plots show the layout of this case, along with the boundary conditions. (Note that particular variations of the BCs at the inflow, top wall, outflow, and side walls may also work and yield similar results for this problem.)

Another important note: although M=0.2 is low enough that the flow is “essentially” incompressible, this is a compressible flow verification case. Therefore, if you run this case with an incompressible code, your results may be close - but not quite the same - as the grid is refined.

## Grid Convergence and results¶

A series of 6 nested 3-D grids are provided. Each coarser grid is exactly every-other-point of the next finer grid, ranging from the finest 65 x 1409 x 641 to the coarsest 3 x 45 x 21 grid. The finest grid has minimum spacing at the wall of y=5 x 10-7, giving an approximate average y+=0.06 over the plate at the Reynolds number run. Even the coarsest grid has reasonably fine wall-normal spacing, giving an approximate average y+=2.0 over the plate (although this grid is probably far too coarse in general to be useful). The grid family is stretched in the wall-normal direction, and is also clustered near the plate leading and trailing edges. The spacing in the spanwise direction is uniform. The following figure shows a portion of the 9 x 177 x 81 grid:

A grid convergence with SA model was performed one time with elsA v3.6.02, results are compared with CFL3D.

## Results of current version¶

Results were performed on 17x353x161 grid.

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

Jameson centered fluxes

Scalar Artificial Dissipation

Gradients calculated on cell centers then corrected on interfaces

Implicit scalar LU-SSOR

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 2 coarse grid(s)
- 1 iterations on coarse grid
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model

Low speed preconditionning

CGNS entries

Iterations number=40000

CFL=15.000000

Harten isentropic correction coefficient (turbulent system)=0.010000

# Academic-3DHemisphereCylinder¶

## Case presentation¶

The purpose here is to provide a test case for a turbulent flow over a smooth body of revolution in 3D. This case is designed primarily for numerical analysis of turbulence model simulations; e.g., convergence properties, effect of order of accuracy, etc.

The geometry is taken from the experimental model studied in AEDC-TR-76-112, 1976 (“An Investigation of Separated Flow About a Hemisphere Cylinder at 0- to 19-Deg Incidence in the Mach Number Range of 0.6 to 1.5” authored by Tsieh, T.).

The following plots show the layout of the hemisphere cylinder grids, along with recommended boundary conditions. Unlike most other test cases on the Turbulence Modeling Resource website, for this particular test case the structured and unstructured grids were created independently of one another. Thus they are not related, other than the fact that the body shape and exit plane are the same. Note that for this test case the outer boundary extents are different: for the structured grid, it is located approximately 20 unit lengths from the body; whereas for the unstructured grid, it is located approximately 10 unit lengths from the body. In the experiment, the Re/ft was 4.2 million, the radius of the hemisphere was 0.5 inches, and the body length was 10 inches. Thus, the Re/(unit grid length) = 0.35 million with a hemisphere radius of 0.5 in the computational grid (i.e., the unit length in the grid is 1 inch). The reference solutions to be provided are computed at M=0.6, angle of attack of 0 degrees. Reference temperature is taken to be 540 degrees Rankine. At the “farfield” boundary, a Riemann invariant BC is employed, with the external boundary assumed to be freestream conditions based on input Mach number.

For computing forces, the reference area of this case is taken to be 10 in2 (for the full 360 degrees), or 5 in2 (for a half-plane computation of 180 degrees).

Although the reference CFD solutions to be provided are for angle of attack of zero, in the first figure below the experimental surface pressure coefficients from AEDC-TR-76-112 are plotted as functions of x for the turbulent flow at angle of attack of 5 degrees. Results are shown for various positions around the azimuth. In the experiment, phi=0 deg corresponds with the “top” or leeside of the body (which faces away from the wind when the body is at angle of attack), and phi=180 deg corresponds with the “bottom” of the body (facing the wind). Approximate results extracted from a figure in the reference at angle of attack of 0 degrees are shown in the second figure below.

## Grid Convergence¶

A series of 6 nested 3-D grids are provided. The grids range from 161 x 289 x 129 (finest half-plane grid) to 6 x 10 x 5 (coarsest half-plane grid)

A grid convergence with SA and Menter SST model was performed one time with elsA v3.6.02, results are compared with CFL3D.

## Results of current version¶

Results were performed on 161x289x129 grid.

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

Roe upwind fluxes

MinMod limiter

Gradients calculated on cell centers then corrected on interfaces

Implicit scalar LU-RELAX

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 1 coarse grid(s)
- 1 iterations on coarse grid
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model

CGNS entries

Iterations number=7000

Harten isentropic correction coefficient=0.050000

CFL=100.000000

Harten isentropic correction coefficient (turbulent system)=0.010000

# Academic-3DSupersonicSquareDuct¶

## Case presentation¶

The purpose here is to provide a validation case for turbulence models. Unlike verification, which seeks to establish that a model has been implemented correctly, validation compares CFD results against data in an effort to establish a model’s ability to reproduce physics. A Fortran program to create nested grids of the same family is provided here if desired. Data are also provided for comparison. For this particular supersonic square duct case (from Davis and Gessner), the data are from experiments.

The experiment utilizes a constant area square duct of height and width D=25.4 mm, with length x/D=50.

The primary focus of this case is to assess turbulence models for internal duct flow with corners. In such cases, turbulent anisotropies can be important because normal stress differences induce flowfield behavior that cannot be captured with models that make use of the Boussinesq assumption.

The reference Mach number is 3.9, and the flow develops over a length of 50 D. As we are interested in comparing data at 50 D, the CFD grid is made slightly longer: 52 D. The walls of the duct are modeled as adiabatic solid walls, although due to symmetry only one quarter of the duct is computed, and symmetry boundary conditions are applied on two boundaries. The following plots shows the layout of this case, along with the boundary conditions. (Note that particular variations of the BCs at the inflow and outflow boundaries may also work and yield similar results for this problem.)

Some of the experimental data for this case will be shown below. The profiles highlighted here are near the downstream end of the duct at x/D=40 and 50, with a particular focus at x/D=50.

The experimental data reference is: Davis, D. O. and Gessner, F. B., “Further Experiments on Supersonic Turbulent Flow Development in a Square Duct,” AIAA Journal, Vol. 27, No. 8, August 1989, pp. 1023-1030, doi:10.2514/3.10216.

Note that skin friction coefficient values are given here based on Uref and rhoref, for convenience when comparing against CFD results. However, the original reference gives Cf based on local edge velocity and edge density. The translation was accomplished using approximate values taken from figs. 8 and 11(b) of the reference. Also note that all of the experimental data provided here were digitized from figures in the reference, so they should be used with caution.

## Grid Convergence and results¶

A series of 6 nested 3-D grids are provided. Each coarser grid is exactly every-other-point of the next finer grid, ranging from the finest 65 x 1409 x 641 to the coarsest 3 x 45 x 21 grid. The finest grid has minimum spacing at the wall of y=5 x 10-7, giving an approximate average y+=0.06 over the plate at the Reynolds number run. Even the coarsest grid has reasonably fine wall-normal spacing, giving an approximate average y+=2.0 over the plate (although this grid is probably far too coarse in general to be useful). The grid family is stretched in the wall-normal direction, and is also clustered near the plate leading and trailing edges. The spacing in the spanwise direction is uniform. The following figure shows a portion of the 9 x 177 x 81 grid:

A grid convergence with SA model was performed one time with elsA v3.6.02, results are compared with CFL3D.

## Results of current version¶

Results were performed on 337x97x97 grid.

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

Roe upwind fluxes

Van Albada limiter

Gradients calculated on cell centers then corrected on interfaces

Implicit scalar LU-SSOR

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 2 coarse grid(s)
- 2 iterations on coarse grid
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model , Quadratic Constitutive Relation, 2000 version

CGNS entries

Iterations number=5000

Harten isentropic correction coefficient=0.010000

CFL=100.000000

Harten isentropic correction coefficient (turbulent system)=0.010000

# Academic-BackwardFacingStep¶

## Case presentation¶

The purpose here is to provide a validation case for turbulence models. Unlike verification, which seeks to establish that a model has been implemented correctly, validation compares CFD results against data in an effort to establish a model’s ability to reproduce physics. A large sequence of nested grids of the same family are provided here if desired. Data are also provided for comparison. For this particular “essentially incompressible” backward facing step problem, the data are from an experiment.

This is a widely-tested configuration. The particular data given here are from Driver and Seegmiller. This is also a test case given in the ERCOFTAC Database (Classic Collection) #C.30 (Backward facing step with inclined opposite wall), and has also been used in turbulence modeling workshops (see references below).

In this case, a turbulent boundary layer encounters a sudden back step, causing flow separation. The flow then reattaches and recovers downstream of the step. The Reynolds number based on boundary layer momentum thickness prior to the step is 5000. This corresponds to a Reynolds number of approximately 36,000 based on step height H. The boundary layer thickness prior to the step is approximately 1.5H.

Boundary conditions appropriate for the CFD are shown in the following figure. Other than a short region with symmetry imposed (to avoid possible incompatibilities between freestream inflow and wall BCs), both bottom and top walls are treated as viscous walls. The original experiment varied the top wall angle, but for this test case here only a straight top wall (zero angle) is considered.

In this case, the inflow length prior to the area of interest (near x=0) has been adjusted so that the naturally developing turbulent boundary layer on the lower wall in the CFD solution grows to approximately the correct thickness and yields approximately the correct wall skin friction coefficient prior to the step. (Note that in some other studies, the inflow boundary layer profile is prescribed near x/H=-4 instead.) The back pressure is adjusted to yield approximately the correct Mach number (M=0.128) upstream of the step. (Note that particular variations of the BCs at the inflow and outflow may also work and yield similar results for this problem.)

Some of the experimental data for this case are shown below. Velocity and turbulence profiles of interest are chosen at the following x/H locations downstream of the step: x/H = 1, 4, 6, and 10.

The experimental data reference is: Driver, D. M. and Seegmiller, H. L., “Features of Reattaching Turbulent Shear Layer in Divergent Channel Flow,” AIAA Journal, Vol. 23, No. 2, Feb 1985, pp. 163-171. See also Eca et al papers: AIAA-2009-3647, AIAA-2007-4089, and AIAA-2005-4728 for summaries of workshops that used this experimental data (although note inconsistencies in ReH).

The Uref is the reference velocity at the center-channel near x/H=-4, used to nondimensionalize velocity and turbulent shear stress profiles. The skin friction coefficient and pressure coefficient data were also both with respect to conditions near this location. Note, however, that when plotted the pressure coefficient data has been shifted uniformly so that Cp is 0 near the position x/H=40 or so. (This was also done by Eca et al in the V&V Workshops referenced above.) The Cp experimental data file provided below contains both original and shifted values.

One of the key measures of success for this flowfield is the prediction of reattachment point downstream of the step. In the experiment, this was determined (by laser oil-flow interferometer measurements of skin-friction and interpolation of the zero skin-friction location) to be:

x/Hreattach = 6.26 +- 0.10

## Results of current version¶

A series of 5 nested 2-D grids, nondimensionalized by the step height H, are provided. As structured grids, these are comprised of 4 zones, connected in a one-to-one fashion (zones 1 and 2 can easily be combined into one zone; and zones 3 and 4 can easily be combined into one zone, if desired). All grid files have been gzipped. Each coarser grid is exactly every-other-point of the next finer grid, ranging from the finest 513 x 513, 193 x 513, 769 x 897, 257 x 897 grid to the coarsest 33 x 33, 13 x 33, 49 x 57, 17 x 57 grid. Note that with these grids the viscous near-wall region on the backside of the step is not resolved to within a y+ of 1. The following figure shows a portion of the grid 3-levels down from the finest grid.

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

AUSMP+ Miles upwind scheme

Third order limiter

Implicit scalar LU-RELAX

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 2 coarse grid(s)
- 2 iterations on coarse grid
- Correction transfer from coarse to fine grid inv_topo
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model

CGNS entries

Iterations number=15000

CFL=5.000000

Harten isentropic correction coefficient (turbulent system)=0.010000

# Academic-FlatPlate¶

## Case presentation¶

The purpose here is to provide a large sequence of nested grids of the same family, along with results from existing CFD codes that employ specific forms of particular turbulence models, in order to help programmers verify their implementations of these same models. On a given grid, there may be differences between the results from different codes, but presumably as the grid is refined the results should approach the same answer (if the flow conditions and boundary conditions are the same). With verification, the purpose is not to establish the “goodness” of a model compared to experiment, but rather to establish that a model has been implemented correctly, as intended according to the equations and boundary conditions. (It is through validation that a model’s “goodness” is established.) Because the purpose here is primarily verification, experiment is not specifically looked at, although law-of-the-wall theory is included for the sake of reference.

The turbulent flat plate case was run at M = 0.2, at a Reynolds number of Re = 5 million based on length “1” of the grid. The body reference length is 2 units. Because the solid wall of the grid extended from x = 0 to x = 2, this means that the Rex at x=1 was 5 million, and Rex at x=2 (the downstream end of the plate) was 10 million. The following plot shows the layout of the simple flat plate grids used for this study, along with the boundary conditions. (Note that particular variations of the BCs at the inflow, top wall, and outflow may also work and yield similar results for this problem.)

Note that for this case the maximum boundary layer thickness is about 0.03 L, so the grid height of y=L is far enough away to have very little influence. For example, a test in which the upper extent was moved down to y=0.48 L only changed results (integrated drag or skin friction at x=0.97) by less than 0.2%.

Another important note: although M=0.2 is low enough that the flow is “essentially” incompressible, this is a compressible flow verification case. Therefore, if you run this case with an incompressible code, your results may be close - but not quite the same - as the grid is refined.

## Grid Convergence¶

A series of 5 nested 2-D grids are provided. Each coarser grid is exactly every-other-point of the next finer grid, ranging from the finest 545 x 385 to the coarsest 35 x 25 grid. The finest grid has minimum spacing at the wall of y=5 x 10-7, giving an approximate average y+=0.1 over the plate at the Reynolds number run. Even the coarsest grid has reasonably fine wall-normal spacing, giving an approximate average y+=1.7 over the plate. The grid is stretched in the wall-normal direction, and is also clustered near the plate leading edge. The following figure shows the 69 x 49 grid (the location x=0.97, where several quantities are later looked at, is pointed out in the figure):

A grid convergence with SA and Kok models was performed one time with elsA v3.6.02, results are compared with CFL3D.

## Results of current version¶

Results were performed on 545x385 grid.

- 2D-Plane configuration
- Steady subsonic flow
- Turbulent viscous flow
- Perfect gas fluid model
- AUSMP+ Miles upwind scheme
- Gradients calculated on cell centers then corrected on interfaces
- Implicit scalar LU-SSOR
- Time integration : Backward Euler
- V-cycle Multigrid convergence acceleration
- Roe upwind flux for transport equations
- Smith (k, l) turbulence model
- Menter turbulence model , Zheng limiter , SST correction
- Earsm-kkl-dmae turbulence model
- Earsm Hellsten turbulence model
- (k, kl) turbulence model
- Spalart-Allmaras turbulence model
- CGNS entries
- Roe upwind fluxes
- Gradients calculated on cell centers
- Iterations number=5000
- CFL=100.000000
- Harten isentropic correction coefficient (turbulent system)=0.001000

# Academic-MixingLayer¶

## Case presentation¶

The purpose here is to provide a validation case for turbulence models. Unlike verification, which seeks to establish that a model has been implemented correctly, validation compares CFD results against data in an effort to establish a model’s ability to reproduce physics. A large sequence of nested grids of the same family are provided here if desired. Data are also provided for comparison. For this particular “essentially incompressible” mixing layer case (from Delville), the data are from experiments.

The experiment utilizes a splitter plate of thickness 3 mm in a tunnel 300 mm wide. The end of the plate is located at x = 0, and the two streams of different velocity fluid merge downstream of this location. The plate is modeled with a taper starting at x = -50 mm, ending up with a trailing edge thickness of 0.3 mm at x = 0.

In the experiment, the upper higher-velocity stream has a boundary layer thickness at x=-10 mm (near the plate trailing edge) of about 9.6 mm and a momentum thickness of about 1 mm. The lower slower-velocity stream has a boundary layer thickness of about 6.3 mm and a momentum thickness of about 0.73 mm. The Reynolds number based on L (= 1 mm, which is length “1” of the grid) is 2900. Reference temperature is Tref = 293 K = 528 R, and speed of sound aref = 343 m/s. If needed, the reference static pressure pref is atmospheric (101325 Pa). The freestream velocities are: 41.54 m/s upper and 22.40 m/s lower.

If applicable to the particular turbulence model being employed, this case should be run with a low freestream turbulence level. In the experiment, Tu was approximately 0.3%.

The upper and lower boundary layer thicknesses are approximated in the CFD by providing different appropriate “run” lengths, and allowing the fully turbulent boundary layers to develop naturally. Isentropic relations are used to obtain inflow boundary conditions in terms of total pressure and total temperature. The upper and lower boundaries are modeled as slip walls, and are contoured slightly in order to yield close to zero streamwise pressure gradient downstream of x=0. The following plot shows the layout of this case, along with the boundary conditions. “Pt” refers to total pressure, “P” refers to static pressure, and “Tt” refers to total temperature.

Some of the experimental data for this case will be shown below. The profiles of interest are at x= 1, 50, 200, 650, and 950, as well as x= -10 on the top and the bottom of the plate.

The experimental data reference is: Delville, J., Bellin, S., Garem, J. H., and Bonnet J. P., “Analysis of Structures in a Turbulent, Plane Mixing Layer by Use of a Pseudo Flow Visualization Method Based on Hot-Wire Anemometry,” in: Advances in Turbulence 2, eds: H.-H. Fernholz and H. E. Fiedler, Proceedings of the Second European Turbulence Conference, Berlin, Aug 30-Sept 2, 1988, Springer Verlag, Berlin, 1989, pp. 251-256.

## Results of current version¶

A series of 5 nested 2-D grids, in units of mm, are provided. As structured grids, these are comprised of 3 zones, connected in a one-to-one fashion. All grid files have been gzipped. Note that there are two different entry boundaries, one above and one below the plate. These two plate lengths are different in order to allow for different boundary layer thicknesses at the end of the plate; the thickness on the top of the plate is larger than the thickness on the bottom. Each coarser grid is exactly every-other-point of the next finer grid, ranging from the finest 737 x 737, 177 x 321, 241 x 321 to the coarsest 47 x 47, 12 x 21, 16 x 21 grid. The following figure shows a portion of the 185 x 185, 45 x 81, 61 x 81 grid (2 levels down from the finest grid). The part of the grid not shown extends to the right to x=1200 mm.

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

AUSMP+ Miles upwind scheme

Third order limiter

Gradients calculated on cell centers then corrected on interfaces

Implicit scalar LU-RELAX

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 2 coarse grid(s)
- 2 iterations on coarse grid
- Correction transfer from coarse to fine grid inv_topo
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model

CGNS entries

Iterations number=40000

CFL=10.000000

Harten isentropic correction coefficient (turbulent system)=0.001000

# Academic-NACA0012-LowSpeed¶

## Case presentation¶

The purpose here is to provide a validation case for turbulence models. Unlike verification, which seeks to establish that a model has been implemented correctly, validation compares CFD results against data in an effort to establish a model’s ability to reproduce physics. A large sequence of nested grids of the same family are provided here if desired. Data are also provided for comparison. For this particular “essentially incompressible” NACA 0012 airfoil case, the data are from experiments.

For the purposes of this validation, the definition of the NACA 0012 airfoil is slightly altered from the original definition so that the airfoil closes at chord=1 with a sharp trailing edge. To do this, the exact NACA 0012 formula

is used to create an airfoil between x=0 and x=1.008930411365 (the T.E. is sharp at this location). Then the airfoil is scaled down by 1.008930411365. Thus, the resulting airfoil is a perfect scaled copy of the 0012, with maximum thickness of approximately 11.894% relative to its chord (the original NACA 0012 has a maximum thickness of 12% relative to its blunted chord, but it, too, has a maximum thickness of 11.894% relative to its chord extended to 1.008930411365). The revised definition is:

Note that the grids supplied in the link below are considered to be appropriate for the level of validation explored here, but are likely not fine enough when high accuracy is required. Convergence properties (“goodness” of results as a function of grid size) are explored in more detail with different grid families in Numerical Analysis of 2D NACA 0012 Airfoil Validation Case. The finer grids on the “Numerical Analysis” page also include the correction in the scaled NACA 0012 formula.

The turbulent NACA 0012 airfoil case should be run at essentially incompressible conditions (the recommendation here is to run M = 0.15 in compressible CFD codes). The Reynolds number per chord is Re = 6 million. Boundary layers should be fully turbulent over most of the airfoil. Inflow conditions for the turbulence variables should be reported. To minimize issues associated with effect of the farfield boundary (which can particularly influence drag and lift levels at high lift conditions), the farfield boundary in the grids provided have been located almost 500 chords away from the airfoil. Otherwise, a “farfield point vortex” boundary condition correction should be employed (see Thomas and Salas, AIAA Journal 24(7):1074-1080, 1986). The following plot shows the layout of the provided NACA 0012 grids, along with typical boundary conditions. (Note that particular variations of the BCs at the farfield boundaries may also work and yield similar results for this problem.)

## Grid Convergence¶

A series of 5 nested 2-D grids are provided. Each coarser grid is exactly every-other-point of the next finer grid, ranging from the finest 1793 x 513 to the coarsest 113 x 33 grid. The finest grid has minimum spacing at the wall of y=4 x 10-7, giving an approximate average y+ between 0.1 and 0.2 over the airfoil at the Reynolds number run. The grid is stretched in the wall-normal direction, and the clustering is maintained in the wake region. The topology is a so-called “C-grid,” with the grid wrapping around the airfoil from downstream farfield, around the lower surface to the upper, then back to the downstream farfield again; the grid connects to itself in a 1-to-1 fashion in the wake. There are 1025 points on the airfoil surface on the finest grid (65 points on the coarsest grid). There are 385 points along the wake from the airfoil trailing edge to the outflow boundary on the finest grid (25 points on the coarsest grid). The figures below show two views of the 449 x 129 grid.

A grid convergence with different models was performed one time with elsA v3.6.02, results are compared with CFL3D.

## Numerical Analysis¶

A series of 7 nested 2-D grids are provided for Families I and II, whereas in the interest of space only the finest grid is provided for Family III. Each coarser grid is exactly every-other-point of the next finer grid, ranging from the finest 7169 x 2049 to the coarsest 113 x 33 grid. (Please contact the page curator if you are unable to extract coarser levels for Family III.) The topology is a so-called “C-grid,” with the grid wrapping around the airfoil from downstream farfield, around the lower surface to the upper, then back to the downstream farfield again; the grid connects to itself in a 1-to-1 fashion in the wake. There are 4097 points on the airfoil surface on the finest grid, with 1537 points along the wake from the airfoil trailing edge to the outflow boundary. The figures below show views of the 449 x 129 grid for each family.

Three families of grids are provided. All have a farfield extent of approximately 500 c. Each family’s finest grid is 7169 x 2049, with minimum spacing at the wall of 1 x 10-7 and average stretching rate normal to the wall of about 1.02 for the points near the wall. The leading edge spacing along the airfoil in all families is the same: 0.0000125 c. The difference between the families is in their trailing edge streamwise spacing:

- Family I: T.E. spacing = 0.000125 c
- Family II: T.E. spacing = 0.0000125 c (10 x finer than Family I)
- Family III: T.E. spacing = 0.0000375 c (3.3333 x finer than Family I, and 3 x coarser than Family II)

Note that none of these grid families is identical to the original family of grids supplied for the NACA 0012 Validation Case. However, Family I should be reasonably similar.

A grid convergence with SA model was performed one time with elsA v3.6.02, results are compared with CFL3D and FUN3D.

## Results of current version¶

Results were performed on 897x257 grid.

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

AUSMP+ Miles upwind scheme

Third order limiter

Gradients calculated on cell centers then corrected on interfaces

Implicit scalar LU-SSOR

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 2 coarse grid(s)
- 2 iterations on coarse grid
- cell2cell_c prolongation type
- Correction transfer from coarse to fine grid inv_topo
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model

CGNS entries

Menter turbulence model , Zheng limiter , SST correction

Iterations number=10000

Boundary treatment of $omega$ in the wall region=linear_extrap

CFL=30.000000

Harten isentropic correction coefficient (turbulent system)=0.010000

# Academic_2D_Wall-Mounted_Hump¶

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

AUSMP+ Miles upwind scheme

Third order limiter

Implicit scalar LU-RELAX

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 1 coarse grid(s)
- 1 iterations on coarse grid
- Correction transfer from coarse to fine grid inv_topo
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model

CGNS entries

Iterations number=40000

CFL=5.000000

Harten isentropic correction coefficient (turbulent system)=0.010000

# Coflowing-Jet-2D¶

Steady subsonic flow

Turbulent viscous flow

Perfect gas fluid model

Roe upwind fluxes

Van Albada limiter

Gradients calculated on cell centers then corrected on interfaces

Implicit scalar LU-SSOR

Time integration : Backward Euler

V-cycle Multigrid convergence acceleration

- 2 coarse grid(s)
- 2 iterations on coarse grid
- Right-term and field restriction synchronous
Roe upwind flux for transport equations

Spalart-Allmaras turbulence model

CGNS entries

Menter turbulence model , SST correction

Iterations number=38000

Harten isentropic correction coefficient=0.050000

CFL=30.000000

Harten isentropic correction coefficient (turbulent system)=0.010000