CFD Benchmarks
1. Introduction
We implement the benchmark proposed by [TurekHron], on the behavior of drag and lift forces of a flow around an object composed by a pole and a bar, see Figure Geometry of the Turek & Hron CFD Benchmark.
The software and the numerical results were initially obtained from [Chabannes].
This benchmark is linked to the TurekHron CSM and TurekHron FSI benchmarks. 
2. Problem Description
We consider a 2D model representative of a laminar incompressible flow around an obstacle. The flow domain, named \(\Omega_f\), is contained into the rectangle \( \lbrack 0,2.5 \rbrack \times \lbrack 0,0.41 \rbrack \). It is characterised, in particular, by its dynamic viscosity \(\mu_f\) and by its density \(\rho_f\). In this case, the fluid material we used is glycerine.
In order to describe the flow, the incompressible NavierStokes model is chosen for this case, define by the conservation of momentum equation and the conservation of mass equation. At them, we add the material constitutive equation, that help us to define \(\boldsymbol{\sigma}_f\)
The goal of this benchmark is to study the behavior of lift forces \(F_L\) and drag forces \(F_D\), with three different fluid dynamics applied on the obstacle, i.e on \(\Gamma_{obst}\), we made rigid by setting specific structure parameters. To simulate these cases, different mean inflow velocities, and thus different Reynolds numbers, will be used.
2.1. Boundary conditions
We set

on \(\Gamma_{in}\), an inflow Dirichlet condition : \( \boldsymbol{u}_f=(v_{in},0) \)

on \(\Gamma_{wall}\) and \(\Gamma_{obst}\), a homogeneous Dirichlet condition : \( \boldsymbol{u}_f=\boldsymbol{0} \)

on \(\Gamma_{out}\), a Neumann condition : \( \boldsymbol{\sigma}_f\boldsymbol{ n }_f=\boldsymbol{0} \)
2.2. Initial conditions
We use a parabolic velocity profile, in order to describe the flow inlet by \( \Gamma_{in} \), which can be express by
where \(\bar{U}\) is the mean inflow velocity.
However, we want to impose a progressive increase of this velocity profile. That’s why we define
With t the time.
Moreover, in this case, there is no source term, so \(f_f\equiv 0\).
3. Inputs
The following table displays the various fixed and variables parameters of this testcase.
Name  Description  Nominal Value  Units  

\(l\) 
elastic structure length 
\(0.35\) 
\(m\) 

\(h\) 
elastic structure height 
\(0.02\) 
\(m\) 

\(r\) 
cylinder radius 
\(0.05\) 
\(m\) 

\(C\) 
cylinder center coordinates 
\((0.2,0.2)\) 
\(m\) 

\(\nu_f\) 
kinematic viscosity 
\(1\times 10^{3}\) 
\(m^2/s\) 

\(\mu_f\) 
dynamic viscosity 
\(1\) 
\(kg/(m \times s)\) 

\(\rho_f\) 
density 
\(1000\) 
\(kg/m^3\) 

\(f_f\) 
source term 
0 
\(kg/(m^3 \times s)\) 

\(\bar{U}\) 
characteristic inflow velocity 

\(m/s\) 
4. Outputs
As defined above, the goal of this benchmark is to measure the drag and lift forces, \(F_D\) and \(F_L\), to control the fluid solver behavior. They can be obtain from
where \(\boldsymbol{n}_f\) the outer unit normal vector from \(\partial \Omega_f\).
5. Discretization
To realize these tests, we made the choice to used \(P_N\)\(P_{N1}\) TaylorHood finite elements, described by [Chabannes], to discretize space. With the time discretization, we use BDF, for Backward Differentation Formulation, schemes at different orders \(q\).
6. Running the case
7. Data files
8. Results
Here are results from the different cases studied in this benchmark.
8.1. CFD1
\(\mathbf{N_{geo}}\)  \(\mathbf{N_{elt}}\)  \(\mathbf{N_{dof}}\)  Drag  Lift 

Reference [TurekHron] 
14.29 
1.119 

1 
9874 
45533 (\(P_2/P_1\)) 
14.217 
1.116 
1 
38094 
173608 (\(P_2/P_1\)) 
14.253 
1.120 
1 
59586 
270867 (\(P_2/P_1\)) 
14.262 
1.119 
2 
7026 
78758 (\(P_3/P_2\)) 
14.263 
1.121 
2 
59650 
660518 (\(P_3/P_2\)) 
14.278 
1.119 
3 
7026 
146057 (\(P_4/P_3\)) 
14.270 
1.120 
3 
59650 
1228831 (\(P_4/P_3\)) 
14.280 
1.119 
All the files used for this case can be found in this rep [geo file, config file, json file]
8.2. CFD2
\(\mathbf{N_{geo}}\)  \(\mathbf{N_{elt}}\)  \(\mathbf{N_{dof}}\)  Drag  Lift 

Reference [TurekHron] 
136.7 
10.53 

1 
7020 
32510 (\(P_2/P_1\)) 
135.33 
10.364 
1 
38094 
173608 (\(P_2/P_1\)) 
136.39 
10.537 
1 
59586 
270867 (\(P_2/P_1\)) 
136.49 
10.531 
2 
7026 
78758 (\(P_3/P_2\)) 
136.67 
10.548 
2 
59650 
660518 (\(P_3/P_2\)) 
136.66 
10.532 
3 
7026 
146057 (\(P_4/P_3\)) 
136.65 
10.539 
3 
59650 
1228831 (\(P_4/P_3\)) 
136.66 
10.533 
All the files used for this case can be found in this rep [geo file, config file, json file]
8.3. CFD3
As CFD3 is timedependent ( from BDF use ), results will be expressed as
where

mean is the average of the min and max values at the last period of oscillations.

amplitude is the difference of the max and the min at the last oscillation.

frequency can be obtain by Fourier analysis on periodic data and retrieve the lowest frequency or by the following formula, if we know the period time T.
\(\mathbf{\Delta t}\)  \(\mathbf{N_{geo}}\)  \(\mathbf{N_{elt}}\)  \(\mathbf{N_{dof}}\)  \(\mathbf{N_{bdf}}\)  Drag  Lift 

0.005 
Reference [TurekHron] 
439.45 ± 5.6183[4.3956] 
−11.893 ± 437.81[4.3956] 
0.01 
1 
8042 
37514 (\(P_2/P_1\)) 
2 
437.47 ± 5.3750[4.3457] 
9.786 ± 437.54[4.3457] 
2 
2334 
26706 (\(P_3/P_2\)) 
2 
439.27 ± 5.1620[4.3457] 
8.887 ± 429.06[4.3457] 

2 
7970 
89790 (\(P_2/P_2\)) 
2 
439.56 ± 5.2335[4.3457] 
11.719 ± 425.81[4.3457] 
0.005 
1 
3509 
39843\((P_3/P_2)\) 
2 
438.24 ± 5.5375[4.3945] 
11.024 ± 433.90[4.3945] 
1 
8042 
90582 (\(P_3/P_2\)) 
2 
439.25 ± 5.6130[4.3945] 
10.988 ± 437.70[4.3945] 

2 
2334 
26706 (\(P_3/P_2\)) 
2 
439.49 ± 5.5985[4.3945] 
10.534 ± 441.02[4.3945] 

2 
7970 
89790 (\(P_3/P_2\)) 
2 
439.71 ± 5.6410[4.3945] 
11.375 ± 438.37[4.3945] 

3 
3499 
73440 (\(P_4/P_3\)) 
3 
439.93 ± 5.8072[4.3945] 
14.511 ± 440.96[4.3945] 

4 
2314 
78168 (\(P_5/P_4\)) 
2 
439.66 ± 5.6412[4.3945] 
11.329 ± 438.93[4.3945] 
0.002 
2 
7942 
89482 (\(P_3/P_2)\) 
2 
439.81 ± 5.7370[4.3945] 
13.730 ± 439.30[4.3945] 
3 
2340 
49389 (\(P_4/P_3\)) 
2 
440.03 ± 5.7321[4.3945] 
13.250 ± 439.64[4.3945] 

3 
2334 
49266 (\(P_4/P_3\)) 
3 
440.06 ± 5.7773[4.3945] 
14.092 ± 440.07[4.3945] 
All the files used for this case can be found in this rep [geo file, config file, json file].
10. Conclusion
The reference results of [TurekHron] have been obtained with a time step \(\Delta t=0.05\). When we compare our results, with the same step and \(\mathrm{BDF}_2\), we observe that they are in accordance with the reference results.
With a larger \(\Delta t\), a discrepancy is observed, in particular for the drag force. It can also be seen at the same time step, with a higher order \(\mathrm{BDF}_n\) ( e.g. \(\mathrm{BDF}_3\) ). This suggests that the couple \(\Delta t=0.05\) and \(\mathrm{BDF}_2\) isn’t enough accurate.
11. Bibliography

[TurekHron] S. Turek and J. Hron, Proposal for numerical benchmarking of fluidstructure interaction between an elastic object and laminar incompressible flow, Lecture Notes in Computational Science and Engineering, 2006.

[Chabannes] Vincent Chabannes, Vers la simulation numérique des écoulements sanguins, Équations aux dérivées partielles [math.AP], Universitée de Grenoble, 2013.