Acoustic pressure of sound generated by the Kirchhoff vortex.
BackgroundNoise pollution has become a great environmental problem. We are interested in the physical understanding of the sound generated by vortices in low Mach number flow and its control. Therefore, we focus on the numerical simulation of vortex sound. Since high accuracy is required over long distances and over long time, accurate and stable numerical methods are needed. Thus, we have developed a high order strictly stable finite difference method for the Euler and Navier-Stokes equations of gas dynamics. Our goal is the accurate numerical simulation of vortex sound and its control. Thereby, we want to contribute to the reduction of the noise generated by vortex shedding for airframes (wing tips, flap side edges, landing gears), rotors (blade tips, blade-vortex interaction), etc.
Euler and Navier-Stokes Equations
Perturbation FormulationWe consider the nonlinear Euler and Navier-Stokes equations of gas dynamics, because we are interested in the noise generation by nonlinear, non-isentropic and viscous effects. Owing to the large disparity of acoustic and mean pressure, density, and other thermodynamic quantities for low Mach number flow, we do not choose the conventional conservative variables as unknowns, but their difference with respect to the stagnation values. Thus, the difference of the thermodynamic quantities becomes a well-conditioned operation and does not suffer from cancellation error when computed in floating point arithmetic.
Entropy SplittingIntroducing entropy variables related to the physical entropy as unknowns, we can express the Euler equations in conservative and in non-conservative symmetric form. Since the conservative variable vector and the flux vector are homogeneous functions of the entropy variables of a degree depending on the choice of the entropy variables, we obtain a canonical splitting or entropy splitting of the Euler equations. The degree determines the conservative and non-conservative fractions of the Euler equations. For this split form of the Euler equations, an energy can be defined. Stability can be proved, if the ingoing characteristic variables are prescribed as boundary conditions. The essential mathematical tool in the stability proof is integration by parts. A more technical requirement is that the eigenvalues at the boundaries can be estimated by an upper bound only depending on the boundary conditions.
Coordinate TransformationFor general geometries, a coordinate transformation from the rectangular 2D or hexahedral 3D computational domain into the physical domain is used. We consider the transformed Euler and Navier-Stokes equations with the perturbed conservative variables as unknowns and the conservative part in perturbation form.
High Order Finite Difference Method
Summation by Parts Difference OperatorHigh order finite difference methods are used which satisfy the discrete analogy of integration by parts called summation by parts (SBP). Pelle Olsson at our department showed that an energy estimate can be obtained for the semi-discrete problem with the spatial derivatives approximated by an SBP difference operator in an analogous way as for the continuous problem. Thus, strict stability of the semi-discrete problem is guaranteed. We have employed a SBP difference operator developed by Bo Strand at our department, which coincides with the standard central sixth-order difference approximation of the first derivative in the interior. Near the boundary, the SBP operator is third-order accurate yielding global fourth-order accuracy. Since the discrete scalar product and norm are based on a positive definite diagonal matrix, the application to multi-dimensions is straightforward. Likewise, the discretization of the metric terms (expressed in conservative form in 3D) and the flow derivatives in the computational domain can be performed with the same SBP difference operator to guarantee freestream preservation of the conservative part. At present, characteristic boundary conditions are injected, i.e. imposed explicity at each time step. Since injection might destroy the SBP property, either a penalty or a projection method will be used to impose physical boundary conditions without violating the SBP principle.
Runge-Kutta MethodThe semi-discretization of the Euler and Navier-Stokes equations yields a large system of ordinary differential equations for the unknown approximations of the perturbed conservative variables at all grid points. We solve the system by the classical fourth-order Runge-Kutta method. The corresponding multistage method for linear problems can be used for almost linear problems. The stability domain of the Runge-Kutta method requires the CFL number to be smaller equal 1.783 for the Euler equations. For the Navier-Stokes equations, the stability condition of both the inviscid and viscous parts has to be respected. Time dependent physical boundary conditions are only imposed after a complete Runge-Kutta time step. At the intermediate stages, the boundary conditions are not imposed but rather the Euler equations are even discretized at the boundary using the SBP difference operator. With this procedure, accuracy is maintained at the expense of a reduced stability.
Characteristic-Based FilterTo suppress spurious oscillations, which cannot be damped by the central interior difference method, we employ a characteristic-based filter after each Runge-Kutta time step. The filter is a simplification of the artificial compressibility method (ACM) filter by H.C. Yee, NASA Ames Research Center, for low Mach number flow. We consider fourth derivatives of the characteristic variables scaled by the corresponding eigenvalues of the flux Jacobian. More accurate filters, which preserve the semi-discrete stability estimate, will be investigated.
Kirchhoff Vortex SoundThe Kirchhoff vortex is an elliptical patch of constant vorticity rotating in irrotational flow. The exact solution of the 2D incompressible Euler equations for an almost circular Kirchhoff vortex is used to prescribe the normal velocity at the circle almost coinciding with the Kirchhoff vortex. Then, the radial acoustic momentum equation yields the boundary condition for the 2D wave equation governing the sound radiated by the Kirchhoff vortex. Because of the periodic angular motion, the 2D wave equation simplifies to the 2D Helmholtz equation. Separation of variables yields the exact solution involving the Hankel function of second-order in the radial direction. The exact solution of the wave equation supplies the exact solution of the linearized compressible Euler equations for the sound generated by an almost circular Kirchhoff vortex.
Numerical ExampleWe consider a Kirchhoff vortex with semi axes deviating only 0.00125 times 2 m from a circle with radius 2 m. It rotates with angular frequency 82.5 1/s. The stagnation density is 1.3 kg/m^3 and the stagnation speed of sound 330 m/s. A polar grid mildly stretched near the Kirchhoff vortex in the radial direction, and uniform in the circumferential direction of 129 times 24 is used. Nonreflecting boundary conditions are used at the farfield. The figures show the acoustic pressure at the time 5/22 s. The quadrupole structure of the acoustic pressure is correctly computed with the high order difference method. The agreement with the analytical solution is excellent.
Acoustic pressure contours of sound generated by the Kirchhoff vortex.
Aeolian TonesAeolian tones generated by vortex shedding from a circular cylinder are simulated using the high order difference method outlined above. We consider a freestream Reynolds number of 150 based on the cylinder diameter and freestream Mach numbers of 0.1 and 0.2. Flow quantities like the Strouhal number, drag and lift coefficients and acoustic quantities like the instantaneous fluctuation pressure are in good agreement with reference solutions. Whereas the unsteady flow can be correctly prediced even with single precision using the perturbation formulation of the Navier-Stokes equations, double precision is required to correctly simulate sound propagation.
Mach number contours of Karman vortex street behind a circular cylinder (Reynolds number 150, Mach number 0.2).
Instantaneous fluctuation pressure contours of aeolian tones generated by vortex shedding from a circular cylinder (Reynolds number 150, Mach number 0.2).
The original publication will be available in LINK at http://link.springer.de. Also published as Technical Report 2001-004, Department of Information Technology, Uppsala University, Febr. 2001, and as RIACS Technical Report 01.02, Jan. 2001, NASA Ames Research Center.
Department of Energy and Process Engineering,
NTNU, Kolbjørn Hejes vei 2, NO-7491 Trondheim, Norway