Viiflow is an aerodynamic analysis method. It enables the fast evaluation of airfoil characteristics, such as lift and drag coefficients or transition locations. It is currently suitable for subsonic cases of single or multi-element airfoils without strong wake confluence or massive separation. It is especially useful for coupled problems, such as fluid-structure interaction or optimization. To cite viiflow, use .
Viiflow comes as a 64bit Cython wheel for Windows and Linux. Cython wheels are installation packages of compiled modules usable from Python. To use it, you therefore need Python (3.7+).
Now, if you are note comfortable already with Python you may want to start here. While no serious programming is necessary to use viiflow, it will be very beneficial if you know how statements and loops work to define your airfoil analysis.
If you have little experience with installing and running Python on Windows: I found it easiest to rely on scoop and use:
scoop install python
For the examples you will also need jupyter, so use
pip install jupyter after installing python.
If you have little experience with Python on Linux: By default, a lot of Linux distributions see Python 2.7 as the default version, though this is changing. Check whether you have Python3.7 installed by using
python3 in a shell. You may need to use
python3 -m pip install jupyter,
python3 -m jupyter notebook for all commands.
To install the wheels, use
pip install viiflow-[...].whl pip install viiflowtools-[...].whl
from a shell.
While viiflow is compiled cython code, it calls numpys linear algebra solve routine at every iteration. For this to be efficient, you may need an efficient numpy installation. Here you can find wheel distributions of numpy for windows built with Intel MKL.
The viiflow library has four methods:
The setup routine can be called with a list of parameters and returns a parameter structure. The call
import viiflow as vf s = vf.setup(Re=1e6,Alpha=10) s.Ma = 0.1
defines the setup structure
s with the parameters Reynolds number 1e6 and angle of attack 10°. After creation, the Mach number was changed from its default value (0) to 0.1. The following parameters can be used. The parameters not available in Basic are only available in the Pro version of viiflow.
|double||Ma||Mach Number (<1)||0||✓|
|double||Ncrit||Critical amplification factor||9||✓|
|int||Itermax||Maximum number of Newton iterations||100||✓|
|bool||IterateWakes||Recalculate wake shape during iterations||0||✓|
|int||Substeps||No of steps between two panel nodes (*1)||1||✓|
|bool||Silent||Do not print info during iterations||0||✓|
|bool||VirtualGradients||Calculate gradients w.r.t. virtual displacement||0||×|
|double||Alpha||Angle of attack||0||✓|
|double||PitchRate||Pitching rate about (0,0) (*2)||0||✓|
|double||LocusA||G-beta constant A (*3)||6.75||✓|
|double||LocusB||G-beta constant B (*3)||0.83||✓|
|double||ViscPwrExp||Viscosity-temperature model exponent||0.7||✓|
|double||WakeLength||Length of all wakes w.r.t. chord.||1||✓|
|int||IncompressibleBL||If>0, calculate boundary layer with Ma=0||0||✓|
|double||StepsizeLimit||Additional limit on Newton step, between 0 and 1||1||✓|
|double||ShearLagLambdaWake||Parameter \lambda in the lag equation (Wake)||.9||✓|
|double||ShearLagLambdaFoil||Parameter \lambda in the lag equation (Airfoil)||1||✓|
|int||HalfWakes||If>0, use two distinct boundary layers for wake||0||✓|
|int||ShearLagType||Defines choice of K_C function in shear lag equation (*4)||0||✓|
|double||TransitionStepLimit||Restrict movement of transition||inf||✓|
During one iteration viiflow marches from the stagnation point along the pressure and suction side of the airfoil to solve the boundary layer equations. The boundary layer equations are essentially a set of ordinary differential equations (ODEs).
By default, the discretization of the ODE on the surface of the airfoil is the same as the panel nodes, i.e. a step along the surface is a single step from panel node to panel node.
By setting substeps to n, the path between to panel nodes is divided into n segments for the ODE solver. This does increase the numerical effort during the march, but not the effort for the much larger problem of the global Newton step.
Pitching airfoils experience different flow conditions than non-pitching airfoils. A pitch rate is given as the physical pitch rate in rad per sec divided by the speed of the free flow. For example, setting pitch_rate to 40*pi/180/20 would be a pitch rate of 40°/s at a current speed of 20m/s. The pitching motion is assumed to be about (0,0), so arrange your geometry accordingly.
Usually the airfoils analyzed will have a chord length of 1m and the results are scaled to the application. To model a pitching airfoil (40°/s) for a model plane at 20 m/s that has a chord length of 7cm where viiflow is calculating the airfoil characteristics at a length of 1m, one would set the pitch rate to 40pi/180/200.07.
The quantity describes as well the chord to radius ratio for rotating airfoils, so set pitch_rate to c/R if given.
The parameters A and B modify the equations used in the turbulent boundary layer. They are part of an empirical correlation called the G-beta locus [2,5] and influence turbulent separation.
Exemplary values from  are : XFOIL default  (A=6.7, B=.75), Boeing (A=6.935, B=.70, reference not found by the author), Green et al. (A=6.43, B=.80), RFOIL  and viiflow defaults (A=6.75, B=.83).
Depending on this integer value, the factor K_c in the shear-lag equation is calculated differently.
For a value of 0 the factor is a constant 5.6. This is the default case.
For a value of 1, the RFOIL K_C function is used, that is
K_C = 4.65-.95*tanh(.275*H-3.5). The shear stress follows the equilibrium value more slowly for large H.
For a value of 2, the K_C function is calculated as described by Ye following Thomas, that is
K_C = 3.75*(3*H)/(H+2). This is similar to the behavior in XFOIL, for high H the shear stress follows the equilibrium faster.
This has a significant effect near and post stall.
The init routine can be called to initialize the variables that are used during the iterations. These are
p, which contains the airfoil geometries, wake geometries, viscid and inviscid solutions and the lift and moment coefficient. E.g.
p.CLreturns the calculated lift coefficient and
p.foils.Xreturns the airfoil geometry as a 2xN ndarray.
bl, which is a list of structures for every airfoil. Every structure, among other things, contains the substructures
bl_fl, the airfoil surface boundary layer, and
bl_wk, the wake boundary layer.
x0, which is the variable used in the Newton iteration in
import viiflowtools.vf_tools as vft # Read Airfoil coordinates into numpy 2x220 array using a function from viiflowtools RAE = vft.repanel(vft.read_selig("RAE2822.dat"),220) # Init takes a list of airfoil geometries, here this list is a single airfoil. # A single 2xN array is fine as well. [p,bl,x] = vf.init([RAE],s) # [p,bl,x] = vf.init(RAE,s) works, too print(p.gamma_inviscid) print(bl.bl_fl.nodes.theta)
The above code read an airfoil into a numpy array, and lets the init function initialize
x. the '
This routine is called to drive the problem towards a solution using
s.itermax Newton iterations.
A very simple call would be
for AOA in range(0,10): s.Alpha = AOA [x,flag] = vf.iter(x,bl,p,s) if flag: # if flag = 1: converged print('AOA %u CL %f CD %f'%(AOA,p.CL,bl.CD))
Above, the call to iter lets it run until convergence or the defined maxiumum number of iterations, overwriting
x in the process. If the iterations were successful, the lift and drag coefficients are printed.
A more advanced call is
res = None grad = None s.Itermax = 0 #No internal iteration for iter in range(100): [x,flag,res,grad] = vf.iter(x,bl,p,s,res,grad) x -= 0.01*np.linalg.solve(grad,res) if np.sqrt(np.dot(res.T,res))<1e-5 print('Now close enough for me!) break
This allows for a fine-grained control of the iterations. Here, the iterations are stopped when the residual falls below
For the usage of gradient information I suggest looking into the Fluid-Structure Interaction example.
This function calculates the velocity at points
X given as numpy 2xN arrays. The calculation is based on the inviscid (panel) method, but includes the displacement body due to the boundary layer. Therefore, it is suitable to estimate the velocities sufficiently far away from the boundary layer.
The function needs a memory buffer where the velocity is written to. The function needs 2xN buffers and positions and for single point use the 2D vectors need to be given to the function as shown in the example.
# Assuming we have a solution (p,bl) from viiflow we generate a single streamline N = 1000 x = np.zeros((2,N)) x[0,0] = 0.1 x[1,0] = 0.1 v = np.zeros((2,1)) for k in range(1,N): vf.inviscid_velocity(p,x[:,k-1:k],v[:,0:1]) # Normalize for streamline integration v/=np.sqrt(v**2+v**2) # Tiny step forward x[:,k] = x[:,k-1] + v[:,0]*1e-3 # Plot with airfoil fig,ax = plt.subplots(1,1) ax.plot(x[0,:],x[1,:]) ax.plot(p.foils.X[0,:],p.foils.X[1,:]) ax.axis('equal')
This function estimates the velocity profile of a single boundary layer node. For laminar boundary layer nodes, a Pohlhausen function is assumed. For turbulent surface boundary layer nodes the explicit Musker function and the Finley wake function as described in  is used. For turbulent wake nodes with low shape factors, a simple power law is assumed. The function takes to 1D buffers for the Y coordinate and the velocity U.
# Assuming we have a solution (p,bl) from viiflow we generate a single streamline # Buffers Y = np.zeros((50)) U = np.zeros((50)) # Calculate profile at trailing edge, pressure side node = bl.bl_fl.nodes[-1] vf.viscid_profile(node,Y,U) # Plot profile fig,ax = plt.subplots(1,1) ax.plot(U,Y) # Plot function in wake vf.viscid_profile(bl.bl_wk.nodes,Y,U) # Plot profile ax.plot(U,Y)
This function uses the function viscid_profile to calculate all velocity profiles in a boundary layer and calculates the normal flow velocities from the (incompressible) equation d/dx u + d/dy v = 0. The returned matrices Y,U and V are centered on the panels, between the nodes.
# Assuming we have a solution (p,bl) N = 50 # Num. of normal elements [Y,U,V] = vf.viscous_flowfield(bl,N)
This routine is called to set a boundary layer to force transition at a given foil-coordinate x location
[p,bl,x] = vf.init([RAE],s) trans_upper = 0.025 # Transition location on suction side trans_lower = 0.4 # Transition location on pressure side for AOA in range(0,10): s.Alpha = AOA if AOA>5: # Only use forced transition if AOA>5 for some reason vf.set_forced_transition(bl,p,[trans_upper],[trans_lower]) [x,flag] = vf.iter(x,bl,p,s) if flag: # if flag = 1: converged print('AOA %u CL %f CD %f'%(AOA,p.CL,bl.CD))
The lists at the third and fourth argument can have as many entries as there are airfoils, or can be empty. If the given transition location is empty or some coordinate not within the airfoil coordinate range, no forced transition occurs. If only forced transition is allowed, set in addition s.ncrit = np.inf.
There exists a Basic version and a Pro version of viiflow. The Basic version does not allow
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 Drela, Mark, and Michael B. Giles. Viscous-inviscid analysis of transonic and low Reynolds number airfoils. AIAA journal 25.10 (1987): 1347-1355.
 Van Rooij, R. P. J. O. M. Modification of the boundary layer calculation in RFOIL for improved airfoil stall prediction. Report IW-96087R TU-Delft, the Netherlands (1996).
 Drela, Mark. Integral boundary layer formulation for blunt trailing edges. 7th Applied Aerodynamics Conference. 1989.
 Francis H. Clauser. Turbulent boundary layers in adverse pressure gradients. Journal of the Aeronautical Sciences 21.2 (1954): 91-108.
 Green, J. E., D. J. Weeks, and J. W. F. Brooman. Prediction of turbulent boundary layers and wakes in compressible flow by a lag-entrainment method. ARC R&M 3791 (1973).
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 van der Horst, Sander, et al. Flow Curvature Effects for VAWT: a Review of Virtual Airfoil Transformations and Implementation in XFOIL. 34th Wind Energy Symposium. 2016.
 Ye, Boyi. The Modeling of Laminar-to-turbulent Transition for Unsteady Integral Boundary Layer Equations with High-order Discontinuous Galerkin Method. Thesis TU-Delft (2015).
 Thomas, J. Integral boundary-layer models for turbulent separated flows. 17th Fluid Dynamics, Plasma Dynamics, and Lasers Conference. 1984.