Implementation of an Enhanced Flux Formulation for Unsteady Navier-Stokes Solutions G. Xia, S. Sardeshmukh, V. Sankaran and C. L. Merkle Purdue University, West Lafayette, IN 47907, USA
[email protected]
1 Introduction Preconditioned time-marching CFD methods have become established as an accurate and efficient framework for all Mach numbers [1]. However, unsteady solution efficiency and accuracy suffer when the combination of low Mach numbers and high Strouhal numbers is encountered, especially in the context of high-fidelity turbulent and/or acoustics problems [1]. To counter this difficulty, an improved discrete formulation that is valid for all steady and unsteady regimes was proposed in an earlier article [2] for structured centraldifferenced algorithms. In this article, we further extend the modified approach to unstructured finite-volume formulations with implicit relaxation and test it for accurately resolving turbulence dynamics using a second-order discretization framework. Implicit upwind methods on unstructured grids usually adopt point- or line-relaxation procedures for the solution of the linear system. However, we find that the enhanced scheme does not guarantee diagonal dominance that is necessary for the stability of these schemes. Consequently, we retain the traditional flux-difference formulation on the left-hand side and employ the modified formulation only on the right-hand side. The inconsistent formulation thereby retains the accuracy benefits of the improved formulation while, at the same time, preserving diagonal dominance for stability. We employ von Neumann stability analysis to confirm the overall stability properties of the scheme and then demonstrate its application for the propagation of Taylor vortices. Finally, we demonstrate the capability of the scheme to capture turbulence dynamics of the decay of isotropic turbulence.
H. Deconinck, E. Dick (eds.), Computational Fluid Dynamics 2006, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-92779-2 19,
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2 Computational Formulation 2.1 Preconditioned Equations For unsteady flows, the governing equations are written in the so-called dualtime form [1]. Γp
∂Qp ∂Q ∂Ei ∂Vi + + =H+ ∂τ ∂t ∂xi ∂xi
(1)
where the τ represents the pseudo-time derivative, introduced purely for numerical purposes. The terms comprising this pseudo-time term are: ρ0p 0 ρT p uj ρ0p ρδjk u j ρT Qp = uj Γp = 0 T h0 ρp − (1 − ρhp ) ρuk ρhT + h0 ρT In the above matrix, h0 is the total enthalpy and the term ρ0p represents the scaled pseudo-property that is responsible for proper conditioning of the system. The definition of this term is given by: ρ0p =
1 ρT (1 − ρhp ) − Vp2 ρhT
(2)
where: Vp = min max V,
k1 V , k2 V · Str , c Remin
(3)
Remin is the minimum cell-Reynolds number, Str is the physical Strouhal number and k1 and k2 are scaling constants. 2.2 Traditional Flux Formulation The traditional flux-difference formulation for an upwind finite-volume scheme takes the following form for the interface flux: ˜ = 1 (EL + ER ) − 1 Γp |Γp−1 Ap |(QpR − QpL ) E (4) 2 2 With appropriate definitions of the left and right states, the above scheme can be viewed as first-, second- or higher-order accurate. Clearly, the second term in the above equation which represents dissipation effects is a function of the preconditioning matrix. As shown in previous research [1], this term becomes overly dissipative for low Mach numbers and high Strouhal numbers. Specifically, the scheme adds too much dissipation in the momentum and energy equations, while adding the appropriate amount needed for stability in the continuity equation.
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2.3 Enhanced Flux Formulation The enhanced flux formulation consists in replacing the offending matrix dissipation term with one that possesses the correct scaling properties. The corresponding flux becomes: ˜ = 1 (EL + ER ) − α Γp |Λ|(QpR − QpL ) E (5) 2 2 where Λ = diag(σ[Γp−1 Ap ], uδkk , u). In other words, the spectral radius of the preconditioned Jacobian is used only in the dissipation terms of the continuity equation, while the standard convective velocity scale is used in the momentum and energy equations. Thus, the offending dissipation terms in the momentum and energy equations are effectively removed in this formulation, while the proper definition is retained in the continuity equation. The enhanced formulation thus insures optimal accuracy for steady, viscous or unsteady computations. 2.4 Linear Solution Procedure The modified dissipation does not guarantee diagonal dominance as required by Gauss-Siedel methods. Indeed, von Neumann stability results (not shown here for brevity) indicate that the scheme is unconditionally unstable in certain Mach-Strouhal regimes. Consequently, we retain the traditional matrix dissipation form on the left-hand side, and employ the enhanced dissipation on the right side alone. A further aspect of this term is the choice of the α parameter. Again, stability theory indicates that the scheme is stable for α ≤ 1. In our current results, we employ α = 0.5.
3 Results 3.1 Taylor Problem To test the artificial dissipation control, we show results from the numerical simulation of the Taylor problem. The Taylor problem represents an analytical solution of the incompressible Navier-Stokes equations for an infinite array of counter-rotating vortices (see Fig. 1), whose amplitudes decay with time: 1 u = − cos x·sin y·e−2νt , v = sin x·cos y·e−2νt , p = − (cos 2x+cos 2y)·e−4νt (6) 4 Figure 2 shows the pseudo-time convergence for several sequential physical time-steps using different schemes. It is evident that the scheme without any preconditioning scaling or only inviscid scaling performs relatively poorly. In particular, the inviscid choice appears to be very stiff for this choice of timescale (CF Lu = 0.1) and performs worse when the time-scale is made smaller. The standard matrix dissipation scheme and the modified dissipation scheme
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both perform very well in comparison, although it is worthwhile pointing out that the modified scheme performs slightly worse, a result of the inconsistent linearization used to insure diagonal dominance.
Fig. 1. Taylor vortex flowfield with periodic boundaries in all directions.
Fig. 2. Pseudo-time convergence for several time-steps of various numerical schemes.
Accuracy of the different schemes is shown in Fig. 3 for two different Reynolds numbers, infinity and Re = 10. The physical time-step in each case is varied so that the viscous time-scale is controlled for the low Reynolds number case. In both cases, it is clear that the modified dissipation scheme provides
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the best comparison with the exact solution and out-performs the traditional matrix dissipation scheme by a significant margin. It is also noteworthy that the no-preconditioning as well as the inviscid-preconditioning cases are both very inaccurate, in part because of convergence issues and in part because of the implications on the sizing of the dissipation terms.
Fig. 3. Comparisons of kinetic energy decay of various schemes for Re = ∞ and Re = 10.
3.2 Decay of Isotropic Turbulence As a second example to test the capabilities of the improved unsteady formulation, we consider the decay of homogeneous turbulence downstream of a grid in uniform flow [3]. The computational results are obtained by the addition of a DES turbulence model to the equations [4, 5]. The results are interpreted as the time evolution of spatially homogeneous turbulence in the time interval determined by the streamwise distance and the mean flow velocity. The computations are initialized using experimental data at a given non-dimensional time and the predictions at a later time are then compared with the experiments. A uniform 64 x 64 x 64 grid is used in the numerical simulation with a mean Mach number of 0.001 and a Reynolds number of 34,000. We note that, since no walls are present, the DES calculation represents a pure LES solution in this case. Figure 4 shows comparisons of the DES predictions with the experimental data for different schemes (left) and for different values of the CDES parameter. It is evident that the modified dissipation scheme provides the best prediction and essentially preserves the accuracy of all the ”represented” modes indicated in the initial data. In contrast, the standard matrix dissipation scheme as well as the other preconditioning choices indicate that the inertial scales are strongly damped. The calibration of the CDES parameter, shown only for the enhanced scheme, indicates that the optimum choice of this parameter is close to 0.5 for this scheme, indicating that the choice of the model parameters are intrinsically tied to the accuracy of the underlying scheme.
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Fig. 4. Comparison of DES predictions with experiments. Left side shows results for different numerical schemes, right-side shows results for different values of CDES .
4 Summary A modified dissipation scheme that promises uniform accuracy at all Mach numbers, Reynolds numbers and Strouhal numbers is presented. In order to preserve diagonal dominance of the implicit linear operator, we retain the traditional flux-difference treatment on the left-hand-side and implement the improved scheme only on the right-hand side. The method is tested with exact results of the Taylor problem and for the case of decay of homogeneous turbulence. Comparisons with the more traditional schemes suggest that the modified scheme is capable of improved accuracy and efficiency, especially for unsteady problems.
References 1. S. Venkateswaran and C. L. Merkle, ’Efficiency and Accuracy Issues in NavierStokes Computations’, AIAA Paper 2000-2251, 2000. 2. Sankaran, V. and Merkle, C. L., ’Artificial Dissipation Control for Unsteady Computations’, AIAA Paper 2003-3695, 2003. 3. Comte-Bellot, G., and Corrsin, S., ’Simple Eulerian Time Correlation of Full- and Narrow-Band Velocity Signals in Grid-Generated Isotropic Turbulence’, Journal of Fluid Mechanics, Vol. 48, pp. 273-337, 1971. 4. Strelets, M., ’Detached Eddy Simulation of Massively Separated Flows’, AIAA Paper 2001-0879, 2001. 5. Xia, G., Sankaran, V., Li, D., and Merkle, C. L., ’Modeling of Turbulent Mixing Layer dynamics in Ultra-High Pressure Flows’, AIAA Paper 2006-3729, 2006.
