Von Karm´ an´ Vortex Streets: Simulating Natural Phenomena at Scale By: Sean Wang April 25th, 2019 1 Introduction The turn of the 19th century ushered in a golden era of fluid and aerodynamics research, with many scientists attacking the great questions that not only arose from advent of the powered airplane, but also those posed by the founders of fluid dynamics decades or centuries before. Possibly the greatest fluid dynamicist of all time, Ludwig Prandtl sought to answer of these questions, particularly as they related to flight forces. In his exploration of bluff body drag in near potential flows, Prandtl observed a natural phenomenon, an oscillatory sheet of shed vortices. This paper will explore this phenomenon as well as its outreaching applications in modelling at scale. 2 The Karm´ an´ Vortex Street The original understanding behind vortex shedding comes from a combination of Prandtl’s boundary layer theory (1904) and the existing potential flow theory. In potential flow theory, for an inviscid, incompressible (∇·U~ ), and irrotational (∇×U~ ) fluid, d’Alembert’s paradox states that there is zero drag. Experimentally, this is never true, with the closest examples being flows of ex- tremely low Reynolds numbers paired with a relatively slow free stream velocity (gen- erally M < :3). In the case of bluff bodies I.E. a cylinder in a flow, there is a strong adverse pressure gradient past the half chord, which causes almost immediate flow separation and a subsequent low pressure wake, integrating across the imbalance of pressures gives the resulting the drag. An additional consequence of separation is the phenomenon discussed earlier, alternating vortices caused by imbalances in the rotat- ing, separating wake; this is what is known as a Karm´ an´ street, after Theodore Van Karm´ an.´ Figure 1: A low pressure wake is visible on a simulated vortex street, a result of separation past the half chord. Re = 800, i = 6000 Von Karm´ an,´ Prandtl’s graduate student proposed idealizations for bluff bodies to predict vortex shedding in 1911, when he assumed that vortices were 2D, equidistant along their respective vortex sheets, and of equal strength whose circulation was given by: I G = Ud~ ~r (1) 1 Von Karm´ an´ went on to further generalize the case of symmetrically staggered vortices of equal circulation (or vorticity). Then the ratio of the distance between vortex sheets b and the longitudinal distance between vortices l can be described: b 1 p = cosh−1 2 ' :2806 (2) l p Von Karm´ an’s´ result is helpful both in predicting and verifying a pattern between vortices for bluff body vortex shedding, but is not an ideal foundation for simulation of such a vortex streets. Instead Navier-Stokes or Lattice-Boltzmann are used to do continuum and/or particle flow simulations. 3 The Lattice-Boltzmann Method The Lattice-Boltzmann method is a subset of fluid dynamics techniques typically used to calculate fluid flows at the microscopic scale. As opposed to macroscopic techniques like Navier-Stokes, which considers the aggregate quantities of a particular flow (den- sity, temperature, velocity), microscopic techniques approach flows as a conglomera- tion of particles instead of a continuum. Figure 2: An example of an octagonal lattice used in LBM. Adapted f rom Bao & Meskas 2011 Lattice-Boltzmann, more specifically, deals with fluids in a discrete lattice, with interactions occurring either at the boundaries or at nodes where particles meet. In this way Lattice-Boltzmann operates very much like gas/fluid lattice automata, with the key factor in vortex street simulation being the streaming and collision processes that occur in the model, defined by: 2 Collision Streaming z }| eq { z }| { [ fi(~x;t) − fi (~x;t)] fi(~x + c~eiDt;t + Dt) − fi(~x;t) = − (3) w eq ~ Where i represents some iterative value, w is a relaxation variable, and fi (x;t) is defined as, eq fi (~x;t) = wir + rsi(~u(~x;t)) (4) where wi are respective weights (0 at the walls: i = 1;2;3), and, e ·~u 9 (e ·~u)2 3 ~u ·~u s (~u) = w (3 i + i − ) (5) i i c 2 c2 2 c2 For a simulation of the Karm´ an´ vortex street, the relaxation factor is taken to be, 1 w = (6) 3nLB + :5 where U rcyl n = ¥ (7) LB Re With the foundations of Lattice-Boltzmann, in equations (3) through (7), a proper simulation can be constructed for the Karm´ an´ vortex street. The following is a code snippet defining non-iterative parameters considered in the vortex street simulation, 1 ###### Flow definition ################################## 2 iterPerFrame = 1000 ’’’Number of iterations before a new 3 frame is saved’’’ 4 maxIter = 50000 # Total number of time iterations 5 Re = 666.0 # Reynolds number :: start @ 220 6 nx = 800; ny = 180; ly=ny −1.0; q = 9 ’’’Lattice dimensions 7 and populations ’’’ 8 cx = nx/4; cy=ny/2; r=ny/9; # Coordinates of the cylinder. 9 uLB = 0 . 0 8 # Velocity in lattice units. 10 nulb = uLB∗ r/Re; omega = 1.0 / (3. ∗ nulb + 0 . 5 ) ; ’ ’ ’ Relax 11 ation parameter’’’ Next, the lattice constants are defined, and the code moves to iterating both stream- ing and colliding over maxIter. Velocities are subsequently stored in an array, then plotted for time intervals based on iterPerFrame (see Page 4, Appendix). 3 Figure 3: The development of the Karm´ an´ vortex street from separation wake, to instability, to periodic vortex street. Re = 800, ULB = :08, iterations ' [6000;46000] 4 4 Non-Dimensionalization for Flow over Volcanic Is- lands In terms of natural phenomena, volcanic islands are particularly interesting because of their particular penchant for shedding Karm´ an´ vortex streets. This should probably be no surprise; volcanic islands are usually isolated enough to encounter relatively uniform free stream flow, and their calderas can operate, at scale, like roughly circular bluff bodies. The Lattice-Boltzmann method can then be used model the vortex street from a given volcanic island. In order to do so, a scaling of non-dimensionalized parameters in the program code is necessary in order to obtain what is expected to be an accurate simulation of these shed vortices. Because Lattice-Boltzmann is sensitive to both high Re and compress- ibility, namely for higher free stream velocities, conservative scaling estimates are pref- erentially chosen to achieve model stability. Important scalings for modelling scale eddies, particularly as they relate to Reynolds number, include the following, r r p = alt (8) rm rSL T U m = ( m )2 (9) Tp Up L r T m = p ( m ):2 (10) Lp rm Tp and Reynold’s number, r U L r U L p p p = m m m (11) mp mm However, for the bulk of volcanic islands, particularly those of interest (alt ' 1800m), altitude variations in r, T, and m are of negligible order of magnitude dif- ference for vortex simulation. The respective proportional scalings are then assumed to be 1. Then comparing absolute scalings, assuming U¥ = 10m=s and LGranCanaria ' 47km, the simulation results can be scaled and compared to that of real observations. Conve- nient Re and ULB are chosen based upon the respective scalings. Then, Re m ULp ReULscale 10 s · 47km 3 = = m ' 587 · 10 (12) ReULm ReULsim (:04) s · 20m The simulation can be run and compare with the expected results. Note from the future: scalings were highly incorrect 5 5 Model Accuracy For Re = 350; ULB = :04, the simulated vortex street is as follows, Figure 4: The development of the Karm´ an´ vortex street from separation wake, to instability, to periodic vortex street. Re = 350, ULB = :04, times scales [10000;18000;20000;49000] 6 The particular island of interest for this simulation is Gran Canaria, in the Canary Island Chain, and is the island for which the scalings were calculated. Composing the simulated vortex street (Re = 350, ULB = :04, i = 49000, see last frame above) and the NASA satellite imagery side by side and marking distances, it is clear that the simulation models the scale vortices of Gran Canaria, quite well. Figure 5: The development of the Karm´ an´ vortex street from separation wake, to instability, to periodic vortex street. Re = 350, ULB = :04, times scales [10000;18000;20000;49000] A further discussion of results of Lattice-Boltzmann is continued on the next page. 7 LBM: Good LBM: Bad Given a correct scaling factor, accu- The calculated scaling factor was com- rately predicts the longitudinal spac- pletely incorrect for length L (probably ing between vortices down the Karm´ an´ my error) street – to a less extent the spacing of Does not accurately predict the long vortices across the street as well region of smooth trailing flow off Gran Simple and relatively fast (though not Canaria before the eddies on Python) Does not give information on relative It’s very nice vortex strengths (useful for anticipat- ing flight hazards) Does not follow von Karm´ an’s´ calcu- b lated l ratio (probably also my error) This particular simulation falls apart at high Re and ULB, a limitation of the code Table 1: A discussion of simulated results and empirical data. 6 Conclusion The phenomenon of Karm´ an´ vortex sheets as a model for scale phenomena, namely eddies off volcanic islands, is not only highly interesting, but carries implications into other applications. It is a theory that could govern or contribute intuition into other such phenomena, an example of which might be trailing tip vortices on commercial airliners, or perhaps the meteorological effects these streets may have. It is an area of fluid/aerodynamics still worthy of exploration today. In simulating these vortex streets, we found the Lattice-Boltzmann Method for fluid flows to be very effective in accurately predicting the geometries of these vortices.