VTM: Vorticity Transport Model

The Vorticity Transport Model is a fully coupled comprehensive rotorcraft analysis code comprised of a Weissinger-L lifting line model for the blade aerodynamics; a non-linear Lagrangian model for the blade dynamics; a vorticity-panel model for the fuselage; and an Eulerian grid based vorticity-velocity CFD model for the rotor wake aerodynamics [1-12]. The VTM solves the unsteady vorticity transport equation which is obtained by taking the curl of the Navier-Stokes equations. Denoting the local flow velocity by u and the associated vorticity distribution by ω, then for inviscid and incompressible 3D flow this equation is stated as follows:

S is a source term representing the vorticity that arises on solid surfaces immersed in the flow. For a rotorcraft, this source arises in the aerodynamic loading on the rotor blades, fuselage, wings, and other parts of the vehicle. The velocity induced by the vorticity distribution at any point in space is given by the solutions to the Biot-Savart relationship,

which, when coupled to Equation 1, feeds back the strength and geometry of the rotor wake into the velocity experienced by the blades and fuselage, and hence to the loading on the helicopter.

The VTM employs a direct numerical solution of Equations 1 and 2 to calculate the evolution of the rotor wake. At the beginning of each timestep the numerical implementation calculates the velocity, u, at which the vorticity field must be advected, by inverting Equation 2 using either cyclic reduction [1, 9, 13] or a Cartesian Fast Multipole method [5, 6, 13]. The vorticity distribution is then advanced through time using a discretized version of Equation 1, obtained using Toro's Weighted Average Flux (WAF) algorithm [14, 15] to reconstruct the vorticity fluxes through cell faces. This process is then repeated for each timestep.

Figure 1: Iso-Surface of Vorticity Showing the Effects of Flux-Limiting on Wake Structure. a) MIN-type Flux limiter. b) SUPER-type Flux limiter [1]

This numerical technique has been shown to preserve the vortical structures of rotor wakes for very long times, since in no way can numerical diffusion and dissipation lead to the destruction of vorticity within the computational domain. Numerical diffusion will lead to the spreading of vorticity, but this can be controlled by implementing a special flux limiter in the WAF scheme. Figure 1 and Figure 2 serve as an illustration of the ability of the VTM wake model to preserve structures in the rotor wake. Figure 1, from Ref. [1], illustrates the effect of the flux limiter used in the WAF algorithm on the structure of the rotor wake. The diagram at left, generated using a MIN-type flux limiter, shows the behavior of traditional methods in dissipating the vorticity in the wake. The SUPER-type flux limiter used in the VTM however, allows the strength of the tip vortices to be maintained throughout the computational domain, as shown in the diagram at right. This has distinct advantages in modeling the interaction between the various components of highly complex rotor configurations, such as that shown in Figure 2 from Ref. [22], where vorticity must be preserved for many rotor revolutions if the accuracy of calculations is to be maintained.

Figure 2: Visualization of the highly interactive flow field around a coaxial helicopter with tail propulsor as calculated by the VTM. [22]

Figure 3, from Ref. [5], serves to illustrate the detail in the wake structure of a hovering rotor that the VTM is capable of capturing. This example serves to show that if the vortical structures in the wake are accurately resolved, then the solution will show experimentally observed fluid dynamic phenomena such as the growth of the vortex pairing instability and the subsequent loss of symmetry of the wake downstream of the rotor.

Figure 3: Wake Visualization of a Two Bladed Hovering Rotor (top) and Four Bladed Rotor in Forward Flight (bottom) showing the development of vortical instabilities downstream of the system [5].

The Vorticity Transport Model has been extensively validated against industry standard test cases such as:
  • Rotor blade flapping motion in low speed forward flight [16] (see Figure 4 left and Refs. [11, 17])
  • Rotor wake trajectory [18] (see Figure 4 right and Ref. [17])
  • Flight in ground effect [19, 20] (see Figure 5 and Refs. [4,23])
  • Fuselage interactions (see Figure 6 and Ref. [21])
For more information on licensing the Vorticity Transport Model contact Glen Whitehouse at CDI 609.538.0444 ext 126.

Figure 4: Comparison of VTM predicted lateral disk tilt with experimental data from Ref [16] and predicted wake structure for a Caradonna and Tung's two bladed hovering rotor (right) [11, 17]

Figure 5: VTM predictions of rotor performance [4] in ground effect (top) and predicted wake structure in various flow regimes [23] (below)

Figure 6: VTM predicted interaction between rotor and fuselage for NASA ROBIN test case [21] (μ=0.15, CT=0.0064). Wake geometry (left), mean out-of-plane inflow (right)


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  21. Kenyon, A.R. and R.E. Brown. Wake Dynamics and Rotor - Fuselage Aerodynamic Interactions. in 63rd Annual Forum of the American Helicopter Society. 2007. Virginia Beach, VA.
  22. Kim, H.W., Kenyon, A.R., Duraisamy, K., Brown, R.E., Interactional Aerodynamics and Acoustics of a Propeller-Augmented Compound Coaxial Helicopter in 9th American Helicopter Society Aeromechanics Specialists' Meeting, Fisherman's Wharf, San Francisco, January, 2008.
  23. Phillips, C., Brown, R.E., Eulerian Simulation of the Fluid Dynamics of Helicopter Brownout, in 64th Annual Forum of the American Helicopter Society, Montréal, Canada, April 29-May 1, 2008.