Saturday, March 1, 2014

Von Karman street part 2

This post is a part of Von Karman Vortex Street tutorial 
Each object invested by airflow generates a wake. Basic air data  instruments can be fooled by wakes, which is directly translated into poor measurement performances. In this mini-series we proceed with a concise presentation of the phenomenon and an introductive tutorial to CFD simulation.  As main example we will use a Von Karman Vortex Street, a good article on the Street dimensions is downloadable here.
Von Karman Street is a phenomenon that can be observed in nature. Below a wake generated by Guadalupe island.
Figure P.35.1 Natural Von Karman Street, Retrieved from this site

In the picture you can see how a constant speed airstream over an island lead to the formation of a periodic vortex wake in the clouds. Our experimental layout will be composed by a simple round bar of radius \(a\) immerged in to an airstream. A bidimensional numerical simulation is carried out; we simulate the plane that sections the bar along the radius plane. The general layout for the simulation is depicted in the following picture.
Figure P.35.2 General simulation layout.

During the simulation the ISA standard conditions are assumed for the air
$$P_{air}=101325Pa$$
$$T_{air}=288,15°K$$
$$\rho_{air}=1,225kg/m^3$$
$$\mu_{air}=1,79e-5Pas$$
$$\gamma=C_p/C_v=1,4$$
$$M_{air}=28,96e-3 kg/mol$$
$$R=8,3145Jmol^{-1}K^{-1}$$
$$R_{air}=\frac {R}{M_{air}}=287,1 Jmol^{-1}K^{-1}$$
The air is modeled as incompressible, that is a fair hypothesis as long as the velocities magnitudes are less than \(0,3M\). Mach number, for ideal gas case, is \(M=\sqrt{\gamma R_{air}T}= 340.3m/s\)
Then 0,3M=102 m/s, we take care to use airspeeds \(U\) well below this value.
Consider the simulation boundaries. At the boundary A, B and D the airflow   is directed along the \(x\) axis and have a known \(U_x\) speed. The constraint at the boundary C is that \(U_y=0\). Along the cylinder wall W the constraint is that speed along x and y axis is equal to zero.
The simulation domain dimensions have impact on the simulation performances and reliability; let's shortly comment how they have been chosen.  If we set the boundaries B and D y velocity component along to zero then we risk to modify the simulation behaviour; we will see how to set alternatively a pressure along the boundaries.  Refer to the following figure that considers only the steady part of a inviscid flow around a cylinder. Our simulation considered the viscosity of the air, nevertheless a comparison with the inviscid analytic solution point out the velocity field modification around the bar.


Figure P.35.3A The streamlines of the flow over a circular cylinder of radius a. An Image linked from here 
Figure P.35.3B Flow over a circular cylinder of radius a. Image from this link

As you note from the precedent figure the component along y axis at an arbitrary distance from the center of the cylinder is not equal to zero. The  eq.11.60  enable us to quantify how far we are from the \(U_y=0\) condition. The flow potential formula along streamlines indicates that we must be  at the infinity to  obtain a zero value for the second term of 11.60, a perturbation in the flow pattern will be perceived all over the plane. Using \(10a\) as y coordinate and\( x=0\) we get a deviation from the ideal case of 1/100. That is a good guess value for the value of distance between the bar center and the top and bottom boundaries. Same reasons yields to choose \(4a\) as the distance between the left boundary and the bar center. If the simulation boundaries are too close to the cylinder the numerical result can be invalidated; look at the following figure, the boundaries are too close and the velocity profile is clearly modified.

P35.4 Boundaries too close to the cylinder, velocity profile is sudden modified next to the boundary

In some occasion, as turnaround, is preferred to simulate a closed geometry and set the B and D boundaries as wall; that is really similar to a wind gallery setup.
Now it’s necessary to choose a value for airstream velocity. Wake behavior change with Reynolds number, have a look to the following figure. The accuracy of shown Reynolds numbers is not relevant.
P35.5 Different wake behaviors in function of Reynolds number. Retrieved here

We chose to go for a Reynolds number of 151 to get a slighty turbolent wake.
Velocity of airstream will be set to \(U_x=\frac{Re\mu}{\rho a}=0,157m/s\)
Periodic wakes have been studied by Strouhal , in particular the vortex formation behind a circular cylinder (Have a look for example to   Blevins, R. D. (1990) Flow-Induced Vibration, 2nd edn., Van Nostrand Reinhold).
Strouhal number is defined as \(Sr=\frac{f_{vortex}a}{U_y}\), it give a relation between frequency of vortex generation and airstream velocity. Strouhal number should be determined by experiments, from bibliography[Blevins, R. D. (1990) Flow-Induced Vibration, 2nd edn., Van Nostrand Reinhold Co, cited also here] at Re=151 Sr should be bounded between 0,15 and 0,22.I take a value equal to 0,18 then \(f_{vortex}=2 Hz\)

In the next post the tutorial about Freecad domain drawing is presented.


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