TY - JOUR

T1 - Preconditioning method for condensate fluid and solid coupling problems in general curvilinear coordinates

AU - Yamamoto, Satoru

N1 - Funding Information:
The authors wish to acknowledge that this study was carried out as a part of the Ground-based Research Announcement for Space Utilization promoted by the Japan Space Forum. Appendix A ˆ i = ρ p U i ∂ ξ i ∂ x 1 ρ ∂ ξ i ∂ x 2 ρ ρ T U i 0 0 0 ∂ ξ i ∂ x 1 + ρ p u 1 U i ∂ ξ i ∂ x 1 ρ u 1 + ρ U i ∂ ξ i ∂ x 2 ρ u 1 ρ T u 1 U i 0 0 0 ∂ ξ i ∂ x 2 + ρ p u 2 U i ∂ ξ i ∂ x 1 ρ u 2 ∂ ξ i ∂ x 2 ρ u 2 + ρ U i ρ T u 2 U i 0 0 0 ρ p HU i ρ u 1 U i + ∂ ξ i ∂ x 1 ρ H ρ u 2 U i + ∂ ξ i ∂ x 2 ρ H ρ T HU i + ρ C p U i 0 0 0 ρ p ( ρ v / ρ ) U i ∂ ξ i ∂ x 1 ρ v ∂ ξ i ∂ x 2 ρ v ρ T ( ρ v / ρ ) U i ρ U i 0 0 ρ p β U i ∂ ξ i ∂ x 1 ρ β ∂ ξ i ∂ x 2 ρ β ρ T β U i 0 ρ U i 0 ρ p nU i ∂ ξ i ∂ x 1 ρ n ∂ ξ i ∂ x 2 ρ n ρ T nU i 0 0 ρ U i , Γ - 1 = ρ C p + ρ T C p T - ρ T φ ρ C p θ + ρ T u 1 ρ T ρ C p θ + ρ T u 2 ρ T ρ C p θ + ρ T - ρ T ρ C p θ + ρ T 0 0 0 - u 1 / ρ 1 / ρ 0 0 0 0 0 - u 2 / ρ 0 1 / ρ 0 0 0 0 1 + φ θ - C p T θ ρ C p θ + ρ T - u 1 θ ρ C p θ + ρ T - u 2 θ ρ C p θ + ρ T θ ρ C p θ + ρ T 0 0 0 - ( ρ v / ρ ) / ρ 0 0 0 1 / ρ 0 0 - β / ρ 0 0 0 0 1 / ρ 0 - n / ρ 0 0 0 0 0 1 / ρ , Γ - 1 A ˆ i = ( ρ C p ρ p + ρ T ) U i ρ C p θ + ρ T ( ∂ ξ i / ∂ x 1 ) ρ 2 C p ρ C p θ + ρ T ( ∂ ξ i / ∂ x 2 ) ρ 2 C p ρ C p θ + ρ T 0 0 0 0 1 ρ ∂ ξ i ∂ x 1 U i 0 0 0 0 0 1 ρ ∂ ξ i ∂ x 2 0 U i 0 0 0 0 ( ρ p - θ ) U i ρ C p θ + ρ T ( ∂ ξ i / ∂ x 1 ) ρ ρ C p θ + ρ T ( ∂ ξ i / ∂ x 2 ) ρ ρ C p θ + ρ T U i 0 0 0 0 0 0 0 U i 0 0 0 0 0 0 0 U i 0 0 0 0 0 0 0 U i , Γ - 1 A ˆ i = L i - 1 Λ i L i , L 1 = 1 0 0 - ρ C p 0 0 0 1 ∂ ξ 1 ∂ x 1 ℓ 1 + ∂ ξ 1 ∂ x 2 ℓ 1 + 0 0 0 0 0 ∂ ξ 1 ∂ x 2 - ∂ ξ 1 ∂ x 1 0 0 0 0 1 ∂ ξ 1 ∂ x 1 ℓ 1 - ∂ ξ 1 ∂ x 2 ℓ 1 - 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 , L 2 = 1 0 0 - ρ C p 0 0 0 0 - ∂ ξ 2 ∂ x 2 ∂ ξ 2 ∂ x 1 0 0 0 0 1 ∂ ξ 1 ∂ x 1 ℓ 2 + ∂ ξ 2 ∂ x 2 ℓ 2 + 0 0 0 0 1 ∂ ξ 2 ∂ x 1 ℓ 2 - ∂ ξ 2 ∂ x 2 ℓ 2 - 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 , Λ 1 = λ ˆ 11 0 0 0 0 0 0 0 λ ˆ 13 0 0 0 0 0 0 0 λ ˆ 11 0 0 0 0 0 0 0 λ ˆ 14 0 0 0 0 0 0 0 λ ˆ 11 0 0 0 0 0 0 0 λ ˆ 11 0 0 0 0 0 0 0 λ ˆ 11 , Λ 2 = λ ˆ 21 0 0 0 0 0 0 0 λ ˆ 21 0 0 0 0 0 0 0 λ ˆ 23 0 0 0 0 0 0 0 λ ˆ 24 0 0 0 0 0 0 0 λ ˆ 21 0 0 0 0 0 0 0 λ ˆ 21 0 0 0 0 0 0 0 λ ˆ 21 , L 1 - 1 = 0 - ℓ 1 - ℓ 1 - - ℓ 1 + 0 ℓ 1 + ℓ 1 - - ℓ 1 + 0 0 0 0 1 g 11 ( ℓ 1 - - ℓ 1 + ) ∂ ξ 1 ∂ x 1 1 g 11 ∂ ξ 1 ∂ x 2 - 1 g 11 ( ℓ 1 - - ℓ 1 + ) ∂ ξ 1 ∂ x 1 0 0 0 0 1 g 11 ( ℓ 1 - - ℓ 1 + ) ∂ ξ 1 ∂ x 2 - 1 - 1 g 11 ( ∂ ξ 1 ∂ x 2 ) 2 ∂ ξ 1 ∂ x 1 - 1 g 11 ( ℓ 1 - - ℓ 1 + ) ∂ ξ 1 ∂ x 2 0 0 0 - 1 ρ C p - ℓ 1 - ρ C p ( ℓ 1 - - ℓ 1 + ) 0 ℓ 1 + ρ C p ( ℓ 1 - - ℓ 1 + ) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 , L 2 - 1 = 0 0 - ℓ 2 - ℓ 2 - - ℓ 2 + ℓ 2 + ℓ 2 - - ℓ 2 + 0 0 0 0 - 1 - 1 g 22 ( ∂ ξ 2 ∂ x 1 ) 2 ∂ ξ 2 ∂ x 2 1 g 22 ( ℓ 2 - - ℓ 2 + ) ∂ ξ 2 ∂ x 1 - 1 g 22 ( ℓ 2 - - ℓ 2 + ) ∂ ξ 2 ∂ x 1 0 0 0 0 1 g 22 ∂ ξ 2 ∂ x 1 1 g 22 ( ℓ 2 - - ℓ 2 + ) ∂ ξ 2 ∂ x 2 - 1 g 22 ( ℓ 2 - - ℓ 2 + ) ∂ ξ 2 ∂ x 2 0 0 0 - 1 ρ C p 0 - ℓ 2 - ρ C p ( ℓ 2 - - ℓ 2 + ) ℓ 2 + ρ C p ( ℓ 2 - - ℓ 2 + ) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 .

PY - 2005/7/20

Y1 - 2005/7/20

N2 - A preconditioned flux-vector splitting (PFVS) scheme in general curvilinear coordinates which can be applied to condensate fluid and solid coupling problems is presented and some typical calculated results are shown to prove the availability of the present method. This method is based on the preconditioning method applied to compressible Navier-Stokes (NS) equations with additional equations and source terms for condensate flows. Since the present PFVS terms composed of the convective and pressure terms of the NS equations are completely reduced to only the pressure terms when the velocities are set to zero, the present scheme can further applied to the calculation not only for a dynamic field but also for a static field including a transitional field from the dynamic region to the static region. In this paper, as a first stage of the present study, coupling problems between a condensate flow in a flow field and heat conduction in a solid structure are simultaneously calculated by using the present method. As numerical examples, transonic and low speed flows around the NACA0012 airfoil, nonequilibrium condensate flows in a nozzle, and natural convection with condensation around a pipe at 1g and zero gravity are simulated with heat conduction in the solid structure.

AB - A preconditioned flux-vector splitting (PFVS) scheme in general curvilinear coordinates which can be applied to condensate fluid and solid coupling problems is presented and some typical calculated results are shown to prove the availability of the present method. This method is based on the preconditioning method applied to compressible Navier-Stokes (NS) equations with additional equations and source terms for condensate flows. Since the present PFVS terms composed of the convective and pressure terms of the NS equations are completely reduced to only the pressure terms when the velocities are set to zero, the present scheme can further applied to the calculation not only for a dynamic field but also for a static field including a transitional field from the dynamic region to the static region. In this paper, as a first stage of the present study, coupling problems between a condensate flow in a flow field and heat conduction in a solid structure are simultaneously calculated by using the present method. As numerical examples, transonic and low speed flows around the NACA0012 airfoil, nonequilibrium condensate flows in a nozzle, and natural convection with condensation around a pipe at 1g and zero gravity are simulated with heat conduction in the solid structure.

KW - Condensate flow

KW - Coupling problem

KW - Heat conduction

KW - Preconditioning method

UR - http://www.scopus.com/inward/record.url?scp=18344378238&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18344378238&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2005.01.013

DO - 10.1016/j.jcp.2005.01.013

M3 - Article

AN - SCOPUS:18344378238

SN - 0021-9991

VL - 207

SP - 240

EP - 260

JO - Journal of Computational Physics

JF - Journal of Computational Physics

IS - 1

ER -