A COUPLED FINITE VOLUME SOLVER FOR THE SOLUTION OF LAMINAR TURBULENT INCOMPRESSIBLE AND COMPRESSIBLE FLOWS

Transcription

1 A COUPLED FINITE VOLUME SOLVER FOR THE SOLUTION OF LAMINAR TURBULENT INCOMPRESSIBLE AND COMPRESSIBLE FLOWS L. Mangani Maschinentechnik CC Fluidmechanik und Hydromaschinen Hochschule Luzern Technik& Architektur Technikumstrasse 21, CH-6048 Horw T , F luca.mangani@hslu.ch C. Bianchini Maschinentechnik CC Fluidmechanik und Hydromaschinen Hochschule Luzern Technik& Architektur Technikumstrasse 21, CH-6048 Horw T , F luca.mangani@hslu.ch

2 Presentation outline Background-State of the Art Coupled Algorithm Developments and present contribution Results and discussion Conclusions 2/18

3 Background-State of the Art The engine of computational fluid dynamics (CFD) is the velocitypressure coupling algorithm that drives the fluid flow In the past years efforts to develop more robust and efficient velocity-pressure algorithms based on: Choice of primitive variables density-based versus pressure-based The type of variable arrangement staggered versus collocated The renewed interest in the development of coupled solvers is due to the increase in computers memory For density but specially for pressure-based algorithms the coupled versus segregated approach dichotomy has not been completely resolved yet! 3/18

4 SIMPLE vs Coupled Algorithm SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) Segregated approach for the pressure velocity coupling Coupled Pressure based U-p coupling without energy Coupled Update properties SIMPLE Update properties Solve continuity, momentum, simultaneously Solve momentum equations Solve turbulence and other scalar equations Update variables Solve pressure-correction (continuity) equation. Update pressure, face mass flow rate Solve energy, turbulence and other scalar equations. Update variables 4/18

5 Coupled algorithm: Momentum Equation Momentum equation discretization The pressure become implicit Pressure goes on the LHS Gradient discretization Coupling coefficients for the momentum equations 5/18

6 Coupled algorithm: Comments If pressure equation is NOT introduced and the momentum and continuity are used: We have a Saddle Block Matrix problem Since no pressure equation is derived, zeros are present in the main diagonal of the discretized continuity equation Leading to an ill conditioned system of equations 6/18

7 Coupled algorithm: Continuity Equation The use of the pressure equation solve the Saddle block matrix solver From the continuity eqaution and momentum we dervied pressure equation Using Rhie-Chow interpolation Discretized equation Coupling coefficients for the pressure equation deriving from fluxes 7/18

8 Matrix Form The final Matrix will be: We choose a cell based variable storage The discretized variables at each control volume are stored together 8/18

9 OpenFOAM Development OpenFOAM cannot handle block matrix ldumatrix dimension is referred to the mesh size Development of a generic interface matrix for external linear solvers Coupled OpenFOAM Sparse matrix PETSc library HYPRE library MUMPS library 9/18

10 Comprimibility and Turbulence If flow is in compressible regime, the change in fluid density should be taken into account The convection flux should also be modified in the pressure equation Turbulence model was added based on k-ω SST model with automatic wall treatment k and ω are solved also in a coupled way A block sparse matrix also for turbulence is solved Coupled U-p Energy Equation Coupled k-ω 10/18

11 Results Comparison is performed between coupled and segregated solver on reference test cases (coupled = solid line, segregated = dash line) Convergence speed is checked plotting normalized 2 residuals ( φ φ ) Uniform initialization, energy and turbulence activated from beginning Incompressible and compressible formulation Inviscid, laminar and turbulent test cases Periodic boundaries Heat transfer R φ = old new ~ ( φold φ ) old 2 11/18

12 GAMM test Inviscid transonic test case Confined circular bump at Ma =0.675 Fixed localco = 600 lower wall upper wall 12/18

13 NACA 012 Isolated 2D profile Inviscid test case Transonic test Ma =0.75,α = 4 Circular domain, inletoutlet BC Fixed localco = 600 Results in terms of dimensionless pressure - C p C p = p p ρ U /18

14 NACA 012 No comparison with segregated due to stability issue related to BC Fast convergence to solution Even though more diffused the shock location is quite well predicted Profile load correctly reproduced 14/18

15 Turbulent flat plate Adiabatic 2D flat plate at Ma in =0.2 Turbulent boundary layer integrated up to the wall y Fixed localco = /18

16 Goldman Test 2D linear cascade -> fully implicit coupled boundary Highly compressible turbulent test case Ma in =0.2 average y + = 50 automatic wall treatment Adiabatic surface, fixed velocity and static pressure Fixed localco = /18

17 Goldman Test Faster convergence than segregated Lower level of residuals than segregated Pressure profile coincident with segregated Good agreement with experimental values Drift respect to other codes due to BC and turbulence model 17/18

18 Conclusions Very fast convergence obtained respect to segregated solvers Implementation of coupling is done arbitrarily on n number of variables Speed of linear solver can be improved Great amount of memory allocation can be reduced with a more efficient implementation Further generalization to be achieved 18/18