An Iterative Approach for Solving the Incompressible Navier-stokes Equations for Simulation of Transition and Turbulence in Complex Geometries

The purpose of this paper is to describe the discrete problem and iterative solution method for spectral simulations of the 3-D Navier-Stokes equations using a general 3-D coordinate mapping. A spectral collocation discretization is proposed along with a multi-level iterative method, and has two unique advantages. First, arbitrary coordinate mappings, which introduce non-constant coefficients into the governing equations, create no additional complication in the spectral collocation approach, but are not feasible for spectral Galerkin and Tau methods. Second, the coupling between velocity and pressure is treated implicitly, allowing either fullyor semiimplicit schemes to be used, and removing the need for the artificial velocity boundary conditions required with fractional-step methods. The performance of the proposed method is competitive with current methods in both memory utilization and operation count. Results were obtained for a canonical case, curved channel flow, to test the suggested approach. Accuracy was examined via the predicted growth rate of a Tollmien-Schlichting wave. Then, simulations for 2-D Dean vortex flow and 3-D breakdown of a saturated Tollmien-Schlichting were obtained. Introduction Direct numerical simulation (DNS) of problems of instability, transition, and turbulence in flows with simple geometric boundaries such as channel, boundary layer, and pipe flows is used extensively to study the complex physical phenomena leading to turbulence. Spectral methods are typically employed for solving the 3-D incompressible Navier-Stokes equations, resulting in an accurate, non-dissipative discretization. For all but the simplest geometry, a coordinate mapping must be used for the computational mesh. The resulting form of the Navier-Stokes equations, however, is not directly amenable to spectral Galerkin or Tau methods, due to the presence of non-constant coefficients, which introduce convolution terms into the wave-number formulation of the governing equations. For this reason, many researchers have turned to spectral element methods which can handle more adequately geometric complexities—see, for example, Guzmán and Amon[1]. Several studies have also been reported in which coordinate mappings have been used along with singleor multi-domain spectral methods. Blodgett[2] has employed a multi-domain spectral collocation method in a 2-D simulation of leading-edge receptivity using a streamfunction-vorticity formulation and a Schwartz-Christoffel (orthogonal) transformation. Carlson et al.[3] have used a time-dependent mapping of the wall-normal coordinate in 3-D DNS in a channel with an emerging obstacle on the lower wall. They used a Fourier Galerkin method in both the streamwise and spanwise directions, and a Chebyshev Tau method in the wall-normal direction. They accomplished this by treating only the Cartesian-like constant coefficient terms in the governing equations implicitly and updating the remaining terms iteratively. The present study proposes an extension to these studies by allowing for an arbitrary 3-D coordinate mapping. The use of collocation, rather than Galerkin or Tau, derivatives allows the implicit part of the algebraic system to include coefficients representing non-constant coordinate metrics. The collocation discretization method ✴ This work is supported in part by AFOSR Grant No. F49620-93-1-0393, and Ohio Supercomputer Center Grant No. PES070-5. is described below. A multi-level iterative approach is then described for solving the resulting nonlinear algebraic system. Finally, 2-D and 3-D simulation results are presented. The Discrete Problem Coordinate Mapping The 3-D incompressible Navier-Stokes equations in non-dimensional form are solved using the skewsymmetric form of the convective terms[4], and are written: ( ) ( ) ∂ ∂ u u u uu u t p + ⋅ ∇ + ∇⋅ = −∇ + ∇ 1 2 1 2 Re , (1) ∇ ⋅ = u 0 . (2) A non-moving general coordinate mapping is employed, and is described as: ξ ξ i i x x x = ( , , ) 1 2 3 , for i = 1,2,3. (3) The governing equations, Eqs. (1) and (2), are written in generalized coordinates by representing vectors and differential operators in terms of covariant and contravariant base vectors associated with the above transformation. See, for example, Gal-Chen and Somerville[5], for a more thorough discussion of generalized coordinates. The base vectors are defined as: Covariant: , Contravariant: , & & e i e i i j i j i i j j x x = = ∂ ∂ξ ∂ξ ∂ (4) for i=1,2,3, and where i j are the Cartesian base vectors. Automatic summation is assumed for repeating indices. The covariant and contravariant base vectors are related to the covariant and contravariant forms of the metric tensor, respectively, and to the Jacobian of transformation for the generalized mapping of Eq. (3), which will be denoted as g . These relationships are: [ ] [ ] ( ) ( ) Covariant: , Contravariant: ,